When dealing with odds, why is the odds getting three 3's on a six sided dice, different depending on if you roll the one die 3 times or roll 3 dice at once?
My favorite Gambler's Fallacy joke: I always take a bomb on flights, because however unlikely it is for there to be a bomb on an airplane, it will be much more unlikely for there to be two.
It may be faulty reasoning, but it actually makes no difference. Whether you fall for the fallacy and bet red, or don't fall for it and bet black, your chances of winning are identical. Falling for the gambler's fallacy is not a recipe for losing. Betting in the first place is the recipe for losing.
I guess technically the change isn't in your chance of winning, but in how you assess the risk. If you think your chances of winning are near certain, you'd be a lot more likely to bet high
It doesn't make a difference in the likelihood of winning/losing. But it makes a huge difference in the willingness of people to gamble in the first place and their decision how much to gamble.
Unless getting the same result repeatedly is evidence of some sort of bias (which I admit is less likely on a roulette wheel, as the colours alternate, and a bias would be more likely to be towards a certain segment of the wheel).
iTunes had this problem when they came out with the shuffle feature many years ago. Users complained that random shuffle wasn’t random enough. There was an expectation that the randomness would result in more or less evenly distributed musical artist for example. Steve Jobs said publicly, we had to change the randomness of the shuffle feature because it was too random.
@Tom R That might have been one thing I bet. I recall that people didn't like getting the same artist too many times in a row, or getting songs from the same album in sequence ... like not expecting 6 heads in a row during a coin flip. They wanted their random shuffle to be more evenly distributed.
I have had 2 cars that could only play cd’s (before play lists were a thing). And I’d put the players on shuffle. One car would randomly play every song on the album before it repeated any song. For the second car, in a span of 7 songs played it could play the same song 3 times. So yeah, the second car was too random.
The flipside of the true random plays, is say on a given cd, theres one particular song you really like, but instead of getting it, you get 3 of the one you like the least. And have to listen to many others 2-3 times before finally getting what you actually wanted. Most turn off that kind of "shuffle". Id expected us to get "1 button press" purchases on car radio music by now, to be replayed at will. Instead its always massive futzing with subscriptions and app stuffs.
@Tom R even that feature can play tracks from the same artist or album in close proximity, albeit of course it’d avoid direct repeats too quickly. Zune would cycle through everything once before repeating for your whole library. Spotify loads in 50 or 100 tracks at a time and won’t repeat within that window (so after a few hours you might get a repetition). I believe at that time iTunes picked a new track randomly from the entire library every single time, such that even going back and forward again yielded a different result (something else Spotify avoids). But after the event mentioned in OP, Apple did add an exclusion list based on the last however-many songs. I don’t think that changed the behaviour of older iPods though, probably to some people’s chagrin.
If a roulette has landed on black 10 times in a row, it would actually have been an indication that the roulette could be mechanically biased in some way, in which case betting on black would actually be better. (Of course, the best move is still to not gamble)
If we assume a casino is very quick to change out a faulty wheel (they had better be), then the 10 blacks in a row becomes less likely to be a mechanical bias, doesn't it?
@@waylonbarrett3456 Even if there is a very small chance that the wheel is bias, bayesian interference can still influence the probability and shift the probability of black, however slightly, above red.
@@thomasfoster1985 I'm not sure what you mean by Bayesian interference. Perhaps, you're referring to Bayesian inference. Anyhow, I don't see the relevance.
@@waylonbarrett3456 He is right. If they change out the wheel then you will see them do it and the new one will no longer be "the wheel that has come up black 10 times in a row", will it?
@@tw9535 My point was that if you find yourself looking at a wheel that has had a large consecutive run, it would probably have the added improbability that it just started acting this way, because if it had been doing this more before you approached, it would have been swapped and you wouldn't be seeing this. So, if you're seeing it, it just started, and it's even rarer than the mere base rate of malfunction. It may even be that you're looking at a well-functioning fair wheel in a rare state. They do happen, after all. Someone has to observe it. Now, I'm not saying casino wheels are ever truly fair. I know they're biased purposefully. When I say fair, I mean adjusted to their desired outcomes.
I recently found this channel and what I like most about it is the presenter’s ability to not only create videos on topics that are of actual interest to me, but also to explain said topics in a manner that is intuitive and easy to understand, primarily through the use of clever examples and well-designed graphics.
When stories of extreme odds come up in relation to Las Vegas people tend to overlook the sheer number of tests being made there. Millions or even tens of millions of hands/rolls/spins per year. Under those circumstances it would be passing strange if you *didn't* get storied instances of major statistical outliers.
The "similar to our experiences" thing gave me a giggle because I remembered an expectation I had as a kid that went the opposite of the example you gave, but still in line with the principle: I was one of three kids, all boys. The first few classmates whose siblings I heard of were similar cases: this girl had a sister, that boy had a brother, and so on, so for a little while I thought that once you had one kid you were locked in and they were all gonna be the same, and when I learned that that's not true I was shocked xD
On introduction of the fallacy at the beginning, I would say that there is one subtle but simple answer to it - Bayesian thinking If I just say a roulette wheel spin Black 5 times in a row, that only happens on a fair wheel about 1 in 30-odd times. This gives a slight, but nonzero probability that there is something *wrong with the wheel*, that it is in fact not fair, but biased towards black somehow. The more consecutive Black that is spun, the more likely it is that there is some mechanical bias in the wheel. At some point, my confidence in this bias exceeds the offered odds of payout and it makes sense for me to bet Black exclusively, from an expected value standpoint. The best part is, that if I am wrong, and the wheel is fair, then it actually doesn't matter which I bet on - if I was going to bet anyway on one or the other, then what I am actually betting on, in a sense, is my own confidence that the wheel is fair or not. Essentially, it challenges the initial assumption that the wheel is fair. The gambler's fallacy accepts the 'fair wheel' as given and derives a wrong conclusion from this due to the way probability works - independent events do not influence each other. But the more information you gain, the more you can legitimately question your confidence in your initial assumptions - every result updates your confidence value in the fairness of the coin up or down, by some amount, even if it is small.
The problem with this thinking is the selection bias used. Why only consider the last 5 spins that were all black? To gather a truly valid view of whether the wheel was biased then all the spins of the wheel ever should be considered.
@DumbledoreMcCracken Neither the Wikipedia article on Bayes nor on his theorem nor Bayesian probability mentions fundamental truth. What example are you referring to? Either way, Roulette wheel spins don't fundamentally differ from coin tosses. What makes you think otherwise?
My brother taught me a strategy he used playing lottery scratchers. He started by asking the cashier questions about the most recent (known) scratchers tickets sold on each ticket roll. First, we find out if the last ticket sold on any roll was a winner, and they are automatically considered Red (No-Play) rolls. Next, we find out which rolls are priming to yield a winner by finding out which roll's last ticket(s) sold were losers, creating the Green (Play) list. For the first round, we purchase one ticket from each roll on our Green list. After scratching each ticket, we take each winner and place their roll on the Red list. In the second round, we purchase one ticket from each roll remaining on our Green list. We repeat with subsequent rounds until each roll hits a winning ticket and no Green rolls remain. This strategy should fall under the Gambler's Fallacy. However, we've maintained average net gains over time. It would be absolutely _fascinating_ to see you evaluate and break down the math in our process.
That’s a sound system in my experience, which at the very least (given the bad overall odds of winning scratch off tickets to begin with) helps me play longer by not losing nearly as much. I do wonder if the gamblers fallacy applies to scratch off tickets as odd/bad as that sounds. Because each new ticket isn’t an independent event like the flip of a coin is. There a guaranteed outcome “known” in advance, prizes are distributed in some logical fashion; ie there aren’t 20 wins in 1 pack and 0 in another.
First of all - The gamblers' mistake was making any bet at all. Roulette is designed to take their money no matter how they play. That said - If I saw a roulette wheel go black 16 times in a row and had to bet, I'd bet on black. 16 in a row might be a sign that the wheel is biased.
Exactly my thought. All the discussion in the video is conditioned on the assumption that the two outcomes are independent and equally probable. 16 blacks in a row would be enough to put the hypothesis into question.
That's an interesting statistical problem in itself - what is the probability that the wheel is biased given that the last 16 goes have all landed on black?
Or worse yet, that the wheel is rigged! Unscrupulous casino owner observes that everybody’s betting on red, and pushes the button that makes the wheel come out black.
@@vigilantcosmicpenguin8721 That's tricky to answer because it depends on the prior probability the wheel was unbiased. Bayes' theorem, IIRC, says that P(A|B)=P(B|A)P(A)/P(B) So, P(Wheeled is unbiased|16 blacks in a rows) = P(16 blacks in a row|wheel is unbiased)P(Wheel is unbiased)/P(16 blacks in a row) It's _doubly tricky_ because P(16 blacks in a row) is itself not independent of whether the wheel is unbiased. Typically the later problem is dealt with via the equation P(B) = P(B|A)+P(B| not A), but this isn't really helpful in this case because it's just as tricky to assign a value to P(16 blacks | wheel is biased), for the simple reason that "wheel is biased" encompasses a _lot_ of possibilities and we'd need to decide on a specific probability model to deal with them. If consider that we're dealing with a continuous probability distribution, and there's no reason we shouldn't, the probability that the wheel is _perfectly_ unbiased, i.e. that the probability of black is _exactly_ 50% is exactly 0, because the probability of _any_ individual value in a continuous probability distribution is 0 (if you aren't familiar with concept, 3blue1brown did a video on it a few years ago called something like "A probability of zero doesn't mean impossible"). In that case, what we'd really need to do is choose some specific margin of error and ask "What's the probability the wheel isn't biased by more than this amount, give 16 blacks in a row?" Or, in other words, "what's the probability that the true probability of black doesn't differ from 50% by more than this amount, given 16 blacks in a row?" This is essentially unwinding all the nuance that classical hypothesis testing hides away in the silly and overly simplistic notion of "statistical significance" that essentially just uses P(B|A) in place of P(A|B). I mean, the actual formal definition of statistical significance doesn't do that per se, but the concept itself really isn't much except to hide away the complexities that are simply inherent to trying to draw conclusions about P(A|B) based on P(B|A), which can be terribly misleading.
They wheel bias is controlled by the spinner person. So I'd have bet against everyone so I would win and the casino would get more money than it paid to me, but only once or twice least the spinner person make it go green
May I suggest, Jade, that you make a video on Monty-Hall problem. It took me good half an hour to figure it out, and it is amazing how many actual mathematicians fell into a trap of its "obviousness" when it was first presented. I would _love_ to see your take on it.
The problem with the monty hall problem is that people assume that monty hall is revealing a goat and offering to switch in order to persuade the player to switch after having selected the car, and believing that had the player selected a goat to start with, the offer would not have been made. It's not like people are allowed to play 10 times in a row
If you run many trials of 10,000 tosses and then looked only at the ones starting with 10 heads in a row you will see that the percentage of heads tends to 5005/10000 = 50.05% not 50%. It does not trend back to 50% but the initial 10 heads just doesn't mean very much in a sample of 10,000.
Does this mean that you can do better than 50% if you bet heads once after seeing 9 heads in a row and then no more bets until 9,990 more rounds have elapsed; then, also wait for another run of 9 and repeat? Why wouldn't this yield better than 50% odds? Would you have found a regularity embedded in the assumed randomness? 🤔
@@waylonbarrett3456 No. If the coin has a 50% chance of heads, it has a 50% chance of heads every time. There is no moment where betting on heads has higher odds than any other moment. That 50.05% that n graner talks about is the percentage including the 10 heads in a row. If you only counted the flips AFTER a streak you would still see 50%.
@@thomasfoster1985 I think you misunderstand what I am asking. This might be a clearer way to make the same point... If 30 blacks in a row is considered rare and 31 blacks in a row is considered rarer, then does it follow that the 31st is more likely to be red? I know that if they are independent, then red is only 50% likely to be the 31st, but then why is 31 black in a row rarer than 30 black in a row? You can't assert a regularity like one pattern being rarer than another and then act like the regularity is not there. The law of large numbers asserts a hierarchy of patterns in large sample sizes. It says 100 heads in a row should be rarer than 2 heads in a row, etc.
@@thomasfoster1985 I believe what n graner is showing is something real about rare events. The transition into and out of the runs seems peculiar and significant.
This also has a big role in sports! Although determining whether events are independent would be pretty challenging I imagine. Small sample sizes are also often used to make dubious judgments about how well a player performed
This is why a good player who succeeds at making low percentage plays no matter how entertaining it is, in the longer run turn out to be a bit disappointing once the luck runs out.
Suppose a team is a on a losing streak, and they believe that a win “is due.” One might say this is falling in the trap of the fallacy, but interestingly believing in the fallacy might actually boost their next performance, increasing the likelihood of a win. I think this seeming paradox is resolved by seeing that - unlike flipping coins - players have memory of their previous games, and so each game is NOT an independent event.
I pondered this paradox when I was younger and realized that any outlier sub-sequence is forever embedded in the overall sequence, and its effect only vanishes after the sequence has so many samples that the outlier is too small to matter. There is no guarantee of any counter-sub-sequence to exactly balance the previous outlier. The chance of such an outlier is the same as the chance of repeating the previous outlier!
Your content is always amazing and very easily accessible to those of us who are mathematically challenged. Thank you. Keep doing a great job my daughter is a fan.
Another way to say this is that after many trials, the statistical influence of a streak diminishes. The Cosmos does not need to revert the frequency to the expected one, it is that the expected frequency changes. Think of it this way. After the first three coin flips are heads, the expected relative frequency for heads for four flips that began with that streak of 3 heads is 87.5%. The expected frequency for 6 trials that begin with a streak of three is 75% heads. Graph the expected frequency of X trials beginning with a streak of three, and when X = 3, the expected frequency is 100%. Then 87.5% for four, etc, until it approaches 50% as X approaches infniity. Graph the actual trials, and the actual frequency will roughly follow that line (curve) down just as it followed the horizontal line when no streak was baked in.
Finally a video where I don't feel like a dropout from elementary school ^^ Though I'd say the fallacy of those gamblers would've been to not bet on the winning colour. Either there is a reason why it spun to black so much like the wheel not being as balanced as everyone would hope or there is none and it doesn't matter what you choose (or there is something you don't know that just increases the odds for the opposite but didn't happen for random chance last time). But *ACTUALLY* the real fallacy was to go into a casino, applying large sample rules and expecting to not loose all their money due to all the games being rigged against the player.
@@One.Zero.One101 yup. I believe only a few configurations for black jack and maybe poker against other players isn't completely rigged against you. But even then, if the casino staff sees you *actually* having figured out something to profit of the games they just throw you out. Don't get me wrong though, I believe there is nothing wrong at going there with a set amount of cash, knowing to go out empty handed, have a fun time, experience the thrill of randomness with stakes and all.
Randomness works in two different ways depending on whether we care about order or not. The probability of "balanced" outcomes (half heads and half tails, for instance) being high is because we often ignore order, so flipping HHHTTT "feels" the same as HTHTHT even though it's a totally different specific order, and there are more possible patterns with 3 heads and 3 tails than, say, 1 head and 5 tails. But each *specific* order is equally probable (in an actually random system). This is shown in Pascal's triangle, where the main numbers in the triangle represent the total number of ways to get to the same "macro-state" where there are the same number of each option. Playing with a quincunx/Galton-board will help folks understand how each different "choice" of going left or right fits into the larger possible set of patterns of specific "micro-states" that Pascal's triangle describes.
I really enjoyed this video. I realize how difficult it must be to constantly come up with new topics and to present them in such an easy to understand way. I salute you for always being up to the challenge.
Presenting them in an easy way is hard, but coming up with the topics is so easy. There are so many fascinating ideas out there in the world! I can't make enough videos for all th amazing ideas!
There's a mathematical/computer science concept called Kolmogorov Complexity that gives a sense of why 20 heads in a row "feels" rare, even though it's the same chance as any other sequence of coin flips. It's one of my favourite mathematical ideas. It's a bit weird to grasp though.
This probably goes beyond the scope of this video, if we're to stay focused on statistics and the gambler's fallacy...but thinking about the laws of large/small numbers, and finding these "clumps" of seemingly ordered results within a large sample of independently random events, which "balance out" over time...it strikes me how the law of large numbers seems to be, if not related, at least oddly similar to entropy. Though I have absolutely no idea if there is any connection at all.
You can see how large number work on a averiges when you try to correct an averige, like averige gas milage by driving more efficient after a period of driving more ‘sportive’. The averige ‘solidifies’ with more trials, is less influenced.
At some point though, you have to wonder if it's more likely that the person who told you that it's a fair coin is lying. 10 heads in a row... plausible. 26 in a row... extremely rare. But after 100 heads in a row, in "real life", the odds are pretty good that the game is rigged.
The most I’ve seen is 13 reds in a row. Lost money betting black until 8 reds showed up. Switched to red after the eighth black and rode the streak figuring the game was slightly rigged. Sometimes you have to bet the hot hand.
There are no fair coins. However, a roulette wheel can have a recognizable bias and still lose you money because the all-lose positions overwhelm the bias.
I don't think it's possible to even make an unfair coin. If it's weighted unevenly, it still spins half and half - it's not going to speed up and slow down between revolutions. The bias in coin flipping i think comes from people who can barely get the thing to turn over when they toss it
We once did the same experiment back in elementary school. A little more than 100 students each rolled a dice 100 times and recorded the results. After taking everything together we got an almost perfect result of 1/6 for each number. I loved this so much since it felt a little bit like magic
Maybe I'm biased.. but I think one of the most important subjects to teach universally is probability and statistics. It's relevant for virtually any inference you'd like to make.. It's nice to see this channel making these ideas mainstream. Excellent coverage of really unintuitive concepts.
I think formal logic would be good to include for the same reasons. It's crazy to me that logic is such a niche subject; I think it should be taught in some way from at least high school and beyond.
I completely agree. An intuitive understanding o multivariate statistics is very useful to understand society and biosciences. I would exchange it for some topics in geometry, trigonometry and logarithms.
Using formal logic would allow you to understand this video is hot garbage. I would honestly love to debate her or anyone on this topic. Flipping a coin 700 million times in a row with the hope it eventually lands on tails is not 1 in 2 probability. Your odds increase with each flip, and this is because the event occurred, not because of some pseudoscientific "memory" of the coin. I've also noticed people who call this a "gambler's fallacy" are most likely politically left wing like NDT, because certain aspects of their brain are either underdeveloped or have a mind numbingly obvious blind spot.
Excellent explanation! The unanswered question for me, though, is when does a small sample set become a large one? If there is a middle ground, then how large is that?
When I was in school, the boys in my crowd liked to match pennies at lunch. My pals always lost to me, because I brought a hundred pennies every day, and they only had three or four each. My large sample size beat their small sample size every time, because I could afford to lose 99 times in a row, while they could only afford to lose four times. This is the real reason why the casino always wins.
Maybe it has some relation to her other video th-cam.com/video/zrFzSwHxiBQ/w-d-xo.html at around 5:45 min in where she explained how kinetic mechanics of a single particle can reconcile with the entropy of the whole system.
This is a great video. On thing I'd have enjoyed as an addendum to it, or a follow-up, is a discussion of the Monty Hall problem - which looks and feels like a gamblers fallacy question, but where the odds actually change with the choice.
An amazing video, Jade! It never ceases to impress me how our cognitive bias, prejudices and believes alter our perception of phenomena, though experimental evidence gives proof enough to question ourselves.
Many events in the natural world are not actually random but chaotic, consider water dripping from a faucet you don’t get one drop right on top of another that would be truly random. So in the natural world there is not independence between actions and the gambler heuristic applies. This would actually be a very interesting topic to do a video on.
@@humanrightsadvocate You know sarcasm is invisible right? Bill didn't DO the study, he paid for it. He pays for a LOT of science research and you are weird.
i made experiment with my son. We counted in a row of flip-coins number of HTHH and compare to number of HHTH and HTTH we found them in different but sometimes overlapping rows. But number of them was almost equal. We made all categorizations that came to our minds. Then i asked him to write 50 results and separately note 2 arrays of real coin-flips and just by taking different type of measurement (for example number of H??TH - with two ignored flips in the middle) i showed him that even when we try then it is hard to do something randomly.
In 10,000 flips, having 200 more heads than tails is just a couple percentage points. In 1000 flips, having 200 more heads than tails is a massive swing.
My area is probability theory and finance, really nice to see this topic covered. A fun comment on the Levy arcsine law is “even the fairest of gambling games will seem unfair most of the time”. Indeed, when we price something like a forward contract in finance, it is expected to only be truly a fair bet at inception, all times after we would expect it to almost always be in one other parties favour. Path dependency is one of those common features that transcends all areas of applied mathematics (all areas of maths). I want to say thank you as my 9 year old asks me all the time what I do and this one (plus the zero knowledge proofs video) let me show him.
When I was in high school, sitting around class bored, I was flipping a coin, calling it in the air, and then scoring whether my call was correct or wrong. On one occasion that was never repeated, I called it correctly 32 times in a row, far exceeding the odds of your roulette wheel example, that's about 4 BILLION to ONE.
I've never understood why you would expect it too "balance out" if anything I would bet on black because the roulette wheel might be broken. The law of large numbers doesn't change anything, if you start at streak of 30 blacks and then do an other 1000000 spins you would expect that on average you see a total of 30 more blacks then reds. The absolute numbers don't balance out, the 30 just becomes negligible to the 1000000
What gets me is that if 30 blacks in a row is considered rare and 31 blacks in a row is considered rarer, then does it follow that the 31st is more likely to be red? I know that if they are independent, then red is only 50% likely to be the 31st, but then why is 31 black in a row rarer than 30 black in a row?
@@waylonbarrett3456 A different way to think about it is: Imagine we setup an incredible number of wheels and operators for those wheels (10^10^10^10 perhaps), and have them all spin 30 times. From here, we tell everyone who has not spun 30 black in a row to leave, congratulate everyone remaining on the incredibly rare thing they just did, and then ask those remaining to spin again. Approximately 50% of those remaining will spin into black again, the rest red. That is to say... 31 black results in a row is just as likely as 30 black results and 1 red result. One is *not* rarer than the other. :) The sequence of 30 black spins in a row is kind of arbitrary. We care about it because it's simple in how recognizable it is. Really, it's not different to how likely you are to be upon the spin after the sequence "bbrrbrbbbbbbrrrrbbbrbbrrrbbrrb", or any other sequence. We only happened to recognize the first sequence because it fits a simple pattern, but that's just doing a disservice to all the other just-as-rare-but-not-simple-pattern-fitting sequences that came before it.
@@ReinOfCats OK, but I was thinking of this all in a frequentist/Bayesian way. Say I'm created 1 billion separate sequences of length 1 million. If I analyze each sequence, I find the probability of 30 consecutive black outcomes and also the probability of 31 consecutive black outcomes. Those numbers are different. Then, I do this with any 2 consecutive run lengths and I wonder why there is this structured hierarchy of probability if the wheel is statistically independent and is not influenced by this zoomed-out order.
This is better than other videos on gambler's fallacy because of how you went into detail about randomness being lumpy on small scales. Very helpful for conceptualising randomness. Thank you.
I have witnessed the gambler's fallacy a lot in the medical world, from both doctors and patients. Anecdotally I've seen the fallacy mean that a patient and doctor have had two very different conversations. Can cause quite a big problem when it comes to surgery and complication rates and such.
Someone once told me they skip every tenth tablet because their doctor had told them that 1 in 10 people who take a tablet get a certain aide effect. Anecdotally, a common lime of thought amongst doctors is 'ive never seen this adverse effect therefore it doesn't exist' which is in a similar realm of misapplying the laws of large/small numbers as the gambler's fallacy
@@robparker1742 Ah, indeed. If you only get 9/10 of the prescribed dose, you can expect to avoid the ⅒ chance of a side effect. Impeccable logic, that.
Most people calculate the probability of a 50/50 outcome sequence incorrectly. Let me explain. In roulette, the probability of Red hitting 4 times in a row is 1/16. However, the probability of four spins producing Red, Red, Red, Black, is also 1/16. The sequence is easier to calculate the probability of when the outcomes are identical, and this is the core issue behind why most gamblers begin to feel as if they can predict the outcome of the next result. It is easy to miss the fact that any string of ten 50/50 outcomes is just as improbable as any other possible 10 iteration sequence.
I enjoyed this video. It reminded me of the time in college 50+ years ago at Lake Sumter Community College in Leesburg, Florida when one of our instructors (Pete Wilson) used a real life experience to demonstrate the law of large numbers. A group of us were carrying a cash box on the way to the gymnasium to sell tickets to some event. The cash box was dropped and coins went everywhere. Pete told us to match coins, one head with one tail, as we picked up the coins. The total should be very close to even... and sure enough it was! This prompted me to do a google search on Pete. I knew that he had gone on to get his PhD and had taught at various top level universities around the country. Sadly I discovered that Pete passed away three years ago. He was a great teacher and I am sure is missed by his friends and family.
2:17 I think you mispronounced the word "adage". I'm not a native English speaker and also because English is not a phonetic language l look up words in an online dictionary and the dictionary pronunciation is different from yours. I don't know if this is because of dialect.
Her pronunciation was close to the French pronunciation (it's a word of French origin). Maybe that's the way they pronounce it in Australia? Or maybe she learned it from her partner who is French?
I’ve had this thought for years: that there seem to be some discrepancy between the law of large numbers and the independence. I’m so glad to see this!!
This is why I get frustrated with time travel shows and movies. If you could memorize the winning lotto numbers and go back in time, there is still the basic probability those numbers will occur. The laws of probability are independent of time. For example, if a person rolled a six sided dice and 2 came out and you went back in time and had the person roll again, there's still in 1 in 6 chance. There's no guarantee a 2 will be rolled. So if you were able to travel 5 years into the past, and then travel back to the future, you will arrive at a different future, since all probability that occurred between those five years would be "re-rolled."
It is worth noting that you described the coincident factors of events taking place as "odds," as in 50/50 or as a percentage and so on, while then noting that these odds do not always seem to stack up over longer sequences. The sequential graphs showing an extended series of events coinciding are, if you think about it, nothing more than a visual representation of the process of arriving at an average of those extended events, where the mathematical "odds" of chance are confirmed by the longer pattern. It would be interesting, for instance, to ask how many throws of a dice have to be made before the mathematical percentage of 6/1 is actually realised, and then to ask if this number of throws is always the same, or whether the same mathematical average can be identified from less or more throws on different occasions, and then to see if there are variables that can affect each the throws on each of these occasions. For me, the point in question is not the odds of a single event, because I see this description as misleading, but how a sequence of single events is formed that gives the average, and how many variations on that sequence there are. Heads and tails are said to be 50/50, but this is a guestimate that asks how many throws of a sequence have to be made to confirm that, and if there are more or less throws to the next confirmation of that same 50/50 statistic, does this highlight variables that are hidden by the pursuit of a 50/50 confirmation.
There is a physical component to consider too, maybe the selection isn't random. Are fish with boils easier to catch? Is the black paint less polished than red?. That is why taking notes in a casino isn't allowed.
"You've had three boys so the next one must be a girl." That's what my uncle thought before having his 2nd, 3rd, 4th, and 5th daughter lol. Or the church family I knew with *eight boys and no girls* in it - although to be fair the first 3 were divided between the parents' respective former marriages. Still for the mother 6 or 7 boys in a row is ridiculous lol
Using ratio of sexes in a family is not really appropriate, because many or even most couples will have some kind of biologically created bias. The bias may be low, but it is not unusual for it to be quite high. So if you come on a family with 5 boys and the mother is pregnant, you really should bet on the next child being male (opposite to the gamblers fallacy) because it's quite likely that you're looking at some kind of systemic organic factor in the relationship between the medical or biochemical character of the two people.
@@realitywave In Germany I was the first to click 👍 So, in a way I have already achieved this lofty goal. Maybe we can say: We were the first in our respective countries?
When I was at university, in an initial physics/physical science course, our professor had been running an experiment in the course for over 20 years. Every class, every day, every semester, students, in lab groups of 4, dropped 100 wooden match sticks, in groups of 10, from a specific, standard height, toward a broad red line in the center of an oblong, metal baking pan. Of course, all the pans, liners, red lines, matches, height, blah blah blah, were identical. The test was to count how many matches from each drop touched the red line. Each lab group tracked their data & reported it to the professor each week. He, in turn, tracked all the data from all the groups for over 20 years, and his graph looked a lot like yours! He demonstrated that, as time progressed, the instances of red line contact approached average - while our individual lab samples were often not average at all.
To me, one of the best things of Breaking Bad was that, having lived in Albuquerque (although I left before the series started), I know those places. I used to go to that car wash. Saul Goodman's office, at least the external set for it, was right next to a stripper joint, which is such a Saul Goodman thing.
Another way to think about it is in terms of macrostates/microstates. Describing the situation of say, "8 heads and 2 tails" when flipping a fair coin ten times is a macrostate, but a particular ordering of that macrostate like "4 heads in a row, followed by 1 tails, followed by 4 heads, followed by one tails" is a microstate; the macrostate description has more microstates (because you can switch up the order) than a macrostate of "10 heads", which is why there is a lower probability of flipping 10 heads in a row, than there is of flipping 8 heads and 2 tails, but any particular microstate has the same probability as flipping 10 heads in a row. The reason why macrostates with more numerically balanced distributions are more probable is because there are a larger number of possible arrangements which suit that description of the overall outcome. So, the reason why you aren't more likely to flip tails after the fair coin lands on heads 9 times in a row is because you are then no longer referring to the probability of flipping 9 heads and 1 tails, but instead, the probability in question is that of 9 heads in a row followed by 1 tails, which is 2^(-10), identically to the probability of getting 10 heads in a row.
1:45 The chances of getting 3 heads in a row is 12,50% while the chances of getting 4 heads in a row is only 6,25%. Wouldn't it be safe to assume that it stops showing head since the likelyhood of this scenario happening is unlikely? For example: the scenario of 10 heads in a row occuring ist around 0.0977%. This scenario is very unlikely to occur, so if I had 9x head in a row I would definitely assume the next one will not be head anymore since I would not think that this 0.0977% is happening right now. I know that in a way it's still just 50/50 and the previous events shouldn't affect the newer ones since they are unrelated in a way... but at the same time I also think there is another factor that noone is thinking about when it comes to this statistics game.
For schools, you can also get strong selection biases. If you look at the SAT scores for schools in different states some states with pretty bad education systems have pretty high average SAT scores. When you look closer at the data though, states with bad education systems don't encourage, or even actively discourage poor students to take the SATs, while other states encourage all students to take them. This brings down those states averages, even if they have better schools overall. A good way to think of the balance between the law of averages and the law of big numbers... say you start flipping a fair coin, and by dumb luck you start with a string of 10 heads, but then suddenly some sort of temporal event causes your timeline to fracture into a million timelines, and you continue, in each of those timelines to flip the coin until you have 10,000 flips in each timeline. On average, at the end of all of those flips, your totals will have 10 more heads than tails, because, on average, each of those timelines will be ruled by the law of averages- but, the law of big numbers will mean that your small fluctuation of +10 heads is a very small portion of the total flips, while the law of averages will mean that some of the timelines will be +- or = to perfect 5k to 5k splits. Of course, the smart thing to do if the wheel keep landing on black (and you are going to be dumb and gamble in the first place) is to watch for any tells like the table manager flipping a switch, and, in the absence of that, maybe give black a couple shots. There is always a chance the table is rigged or no longer fair for some reason, in which case you might as well get in on it (well, depending on your feelings about ripping off a casino).
Strictly speaking, the law of large numbers only applies as the sample size approaches infinity. Also, the word „relative“ does a whole lot of heavy lifting in „relative frequency“. For instance, if you integrate over accelerometer values in order to find out your velocity, the noise adds up rather than averaging out. You‘d need to average the values, but then you don’t get the velocity anymore.
In the opening clip, it might be the case that the roulette table was biased. Apparently, through repeated wear and years of use, the ball sometimes prefers to land on some spaces compared to others. Maybe falling for the gamblers fallacy of "It fell on black so it will continue falling on black" and always betting on black would be profitable assuming that the probability of landing on black is greater than 1/2 (due to the wear).
Which is why scientist have to "publish" your findings... Does multiple things... One of which is "showing the data" which allows others to review how samples were taken and the scope of the data collection that support or refute those finds... In other words, testing how good the tests used were! I have never been disappointed in a single one of your videos! Between that glowing optimistic smile and the solid, well crafted subject matters covered by very clear and concise breakdowns - making each topic manageable and relatable! Thank you.
Thank you so much for this video. When I heard of Gambler's Fallacy and of the fact that as you approach infinite trials the result approaches the probability I was highly confused. It seemed contradictory at the time.
It's been about 7 years since we talked about the law of large numbers in school and this has always been something that I was wondering about ever since.
The movie Jerry and Marge go large talks about this. If you find a flaw in any kind of betting you can’t take advantage of that with a small number of bets. You have to place enough bets for the law of large numbers to kick in. Great overview. It helped me think of it more deeply.
My experience with a gambler , who's attitude seemed to be : something will happen or it won't happen, so on every spin on the slot a jackpot will "hit" or it won't "hit". No math required. I suppose calling a fallacy on any attitude or system a gambler has is contingent on a winning or losing outcome. (me being subjective). Your presentation on "Representativeness Heuristics" reminded me of a Star Trek episode , "The Arena". Kirk and a Gorn are forced to battle each other by a higher entity to settle their dispute. Kirk finds (his initial Gorn "profile") the Gorn, with it's reptilian features grotesque and incorrigible (heuristics?) . Although Kirk triumphs in battle over the Gorn , he decides not to kill it and asks the higher entity not to destroy it either. The higher entity is impressed with Kirk for showing the "advanced concept" of mercy, and express's "there still might be hope for the human race".
My uncle Whizzo was a sucker for the gambler's fallacy. He lost a pile of money wagering on professional tic-tac-toe, he was always convinced that somebody's gotta win eventually.
I like Feynman’s quote very much about this. I had this demonstrated to me in school with coin flips and noting the results down, and she said if we were here all day the numbers would probably get close to 50/50, but we can’t say when any of those balancing moments would occur and it might oscillate or might look one-sided all the way (as born out by your simulations). I thought that did a pretty good job explaining how a thing can be evenly fair overall but randomness is more about when in the overall sequence we witness. Are we in a run of heads time or a run of tails time? No way to find out except to sample it.
A couple of thoughts: Firstly, while the relative frequencies do tend to converge on 50%, the absolute frequencies tend to diverge. After ten thousand coin tosses, you'd expect either heads or tails to be about 100 ahead of the other. For a fair coin, the expected lead is the square root of the number of tosses, so the "error" in the relative value is roughly the square root divided by the number of tosses, or one over the square root (well, half that because if heads is ahead, half the lead is excess heads; half is a shortage of tails). Secondly, one possible contributor to the gambler's fallacy is that there is a similar scenario where red and black are guaranteed to even out in absolute terms, not just relative - and that's drawing (without replacement) from a (well-shuffled) pack of cards. Each individual card drawn is equally likely to be red or black, but drawing a black card makes the next card more likely to be red, and vice versa because you have to end up with twenty six of each colour when you reach the end of the deck...
@@UdoLattek02 Yes, that's why it behaves differently, but for someone just thinking casually about how it would turn out, the two look pretty similar (particularly if you only look at one or two draws/spins)
Brains use bayesian inference which is a good way to think about casually connected events. It's misapplication with independent events is the basis of the gamblers fallacy.
This reminded me of when my coworker told me how they pick lottery numbers. They look at past results for the numbers that have come up most often in the past. And then specifically picks those ones because they're clearly more likely to come up. But then, on the flip side. They also make sure their specific group of 6 numbers has never won before because, "The same set of numbers will never come up twice". I found this strange because not only are we assuming the rolls have memory, we're assuming independently, they're weighted, but in sets, they're due. Obviously both strategies are incorrect and don't help you win. But since the odds of any set of numbers is the same anyway, at least their strategy is fun to figure out and doesn't actively hurt you to do. Unrelated to the video, they also refuse to take numbers in sequence. And always buy two tickets only 1 number off because, "you always get a ticket and miss by like 1 number!"
so with a roulette wheel there is a bunch of factors that could lower randomness, friction, wheels levelness, the human factor, and if you see a pattern emerge there could be a reason for it
Say in the first 10 flips you got 9 heads and 1 tail so your frequency of heads is 90%. But then in the next 2N flips (where N is large) you expect roughly N heads and N tails so your frequency of heads becomes (N+9)/(2N+10). If you graph this function you'll see it starts at 90% but gets closer and closer to 50% as N increases. So the coin doesn't have to be biased in favor of tails after the first 10 flips for the observed frequency to revert to the expected frequency. (See also: regression to the mean)
An interesting corollary is: given a run of tails, say, with what likelihood can we reject the null hypothesis that the coin is fair? If you flip thirty tails in a row in fifty trials, you cannot blithely invoke the gambler's fallacy and say the next flip is 50/50, because it is reasonable to ask the probability of observing that outcome by chance. If the likelihood is too low (for a given value of low) we reject the null hypothesis that the coin is fair. This is when Bayesian statistics enters the picture. Note this is different from asking what the likelihood is of flipping 30 tails in fifty trials - that's not really all that unlikely.
I just rememberd something really fitting: At some point a Phone-Company that i do not like, implemented a random shuffle system in their Music service, but many many Consumers outraged at them telling them the shuffel is not random enough because many songs got playd multiple times at a row. So in the end that "Phone company i do not like" decided to make their already random shuffel less random, to please the crowd into thinking its more random.
There is a difference between chance and probability. Chance in that instance is 50/50, but probability as bounded by sequence length. We tend to look at things in small snapshots of time rather than over longer sequences of time and larger numbers of trials.
I don't know if you've upgraded your lighting setup or I've not been paying as much attention in your previous videos, but this video is ✨ wonderfully lit ✨
I think a nice visual would be showing the same experiment (graph of coin flips/dice rolls) that already starts with 100 heads or 100 6’s. It would also converge to 50% and 1/6 respectively, even with the initial “lucky” streak.
I often wonder why I was forced to take statistics to get a college degree when in my jobs managers have ignored anything in which I used statistics. I once showed that based on failures in previous products, our lack of testing indicated that we would have a more than fifty percent of having a major bug in a product we were about to release. The managers ignored me and released it but we got lucky that time. My boss send out an email saying that since there were no major bugs, my "silly mathematics" were "wrong".
You can't assume that everything that's supposed to be random actually is. Roulette wheels can be fixed, if the bettors are making lopsided bets on red or black then a dishonest croupier can simply select the other color. In the case of a casino where there was an excessively long streak of black while the majority of the bettors picked red the smart thing to do would be to pick black. If the wheel is honest then your color choice doesn't matter, you'll win just under 50% of the time if you stay long enough, the house edge is the 0 in Monte Carlo, and 0 and 00 in Las Vegas. However if the wheel is fixed then you should bet against the crowd. If everyone else is picking red then bet on black. If the crowd switches to black then switch to red.
The thought that occurred to me watching this video was that the entire field of Statistics is based on the Gambler's Fallacy in reverse. The basic premise is that you take a representative sample and then based on the relative sizes of the sample vs the total population work out how likely it is that the sample represents the mean of the total population. There is a little more to it than that, but that is the basics of it. Which just reinforces the cognitive disonance of the Gambler's Fallacy when one doesn't stop to acknowledge the fact the events are independent random events.
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They don’t use “next generation encryption”. They use the same (current generation) encryption as everyone else, as they should.
Better Call Saul is as good if not better. Thank you for your awesome videos!
Before you click aw...click
I got a total brain-crush watching this.... SUBSCRIBED!
When dealing with odds, why is the odds getting three 3's on a six sided dice, different depending on if you roll the one die 3 times or roll 3 dice at once?
My favorite Gambler's Fallacy joke:
I always take a bomb on flights, because however unlikely it is for there to be a bomb on an airplane, it will be much more unlikely for there to be two.
mind boggling
All Fun and games until we catch the same flight
As of today I'll try to find out if there is a "Paul Rimmer" on my flight.
Joking about that at an author could get you in trouble.
To make it clearer you're joking use a box of snakes instead! 🐍 👍
Of course, if everyone thought of that, it would make it even less true than it already is.
It may be faulty reasoning, but it actually makes no difference. Whether you fall for the fallacy and bet red, or don't fall for it and bet black, your chances of winning are identical. Falling for the gambler's fallacy is not a recipe for losing. Betting in the first place is the recipe for losing.
I hit a million on a scratch off ticket 3 times in a row, then had 3 car wrecks in a row losing a limb in each one. Wow what luck.
I guess technically the change isn't in your chance of winning, but in how you assess the risk. If you think your chances of winning are near certain, you'd be a lot more likely to bet high
It doesn't make a difference in the likelihood of winning/losing. But it makes a huge difference in the willingness of people to gamble in the first place and their decision how much to gamble.
Unless getting the same result repeatedly is evidence of some sort of bias (which I admit is less likely on a roulette wheel, as the colours alternate, and a bias would be more likely to be towards a certain segment of the wheel).
The issue is if you fall for the fallacy and bet more than you would have otherwise do.
iTunes had this problem when they came out with the shuffle feature many years ago. Users complained that random shuffle wasn’t random enough. There was an expectation that the randomness would result in more or less evenly distributed musical artist for example. Steve Jobs said publicly, we had to change the randomness of the shuffle feature because it was too random.
@Tom R That might have been one thing I bet. I recall that people didn't like getting the same artist too many times in a row, or getting songs from the same album in sequence ... like not expecting 6 heads in a row during a coin flip. They wanted their random shuffle to be more evenly distributed.
I have had 2 cars that could only play cd’s (before play lists were a thing). And I’d put the players on shuffle. One car would randomly play every song on the album before it repeated any song. For the second car, in a span of 7 songs played it could play the same song 3 times. So yeah, the second car was too random.
@Tom R basically:
people want high entropy, not random entropy
The flipside of the true random plays, is say on a given cd, theres one particular song you really like, but instead of getting it, you get 3 of the one you like the least. And have to listen to many others 2-3 times before finally getting what you actually wanted. Most turn off that kind of "shuffle". Id expected us to get "1 button press" purchases on car radio music by now, to be replayed at will. Instead its always massive futzing with subscriptions and app stuffs.
@Tom R even that feature can play tracks from the same artist or album in close proximity, albeit of course it’d avoid direct repeats too quickly.
Zune would cycle through everything once before repeating for your whole library. Spotify loads in 50 or 100 tracks at a time and won’t repeat within that window (so after a few hours you might get a repetition).
I believe at that time iTunes picked a new track randomly from the entire library every single time, such that even going back and forward again yielded a different result (something else Spotify avoids).
But after the event mentioned in OP, Apple did add an exclusion list based on the last however-many songs. I don’t think that changed the behaviour of older iPods though, probably to some people’s chagrin.
If a roulette has landed on black 10 times in a row, it would actually have been an indication that the roulette could be mechanically biased in some way, in which case betting on black would actually be better. (Of course, the best move is still to not gamble)
If we assume a casino is very quick to change out a faulty wheel (they had better be), then the 10 blacks in a row becomes less likely to be a mechanical bias, doesn't it?
@@waylonbarrett3456 Even if there is a very small chance that the wheel is bias, bayesian interference can still influence the probability and shift the probability of black, however slightly, above red.
@@thomasfoster1985 I'm not sure what you mean by Bayesian interference. Perhaps, you're referring to Bayesian inference. Anyhow, I don't see the relevance.
@@waylonbarrett3456 He is right.
If they change out the wheel then you will see them do it and the new one will no longer be "the wheel that has come up black 10 times in a row", will it?
@@tw9535 My point was that if you find yourself looking at a wheel that has had a large consecutive run, it would probably have the added improbability that it just started acting this way, because if it had been doing this more before you approached, it would have been swapped and you wouldn't be seeing this. So, if you're seeing it, it just started, and it's even rarer than the mere base rate of malfunction. It may even be that you're looking at a well-functioning fair wheel in a rare state. They do happen, after all. Someone has to observe it. Now, I'm not saying casino wheels are ever truly fair. I know they're biased purposefully. When I say fair, I mean adjusted to their desired outcomes.
I recently found this channel and what I like most about it is the presenter’s ability to not only create videos on topics that are of actual interest to me, but also to explain said topics in a manner that is intuitive and easy to understand, primarily through the use of clever examples and well-designed graphics.
Thanks! :)
I can't agree more! She's passionate, articulate, and extremely charismatic. A wonderful presenter.
actually, i am just here for the presentation, every second of it....i LUVE FALLACIES!!!
When stories of extreme odds come up in relation to Las Vegas people tend to overlook the sheer number of tests being made there. Millions or even tens of millions of hands/rolls/spins per year. Under those circumstances it would be passing strange if you *didn't* get storied instances of major statistical outliers.
The "similar to our experiences" thing gave me a giggle because I remembered an expectation I had as a kid that went the opposite of the example you gave, but still in line with the principle: I was one of three kids, all boys. The first few classmates whose siblings I heard of were similar cases: this girl had a sister, that boy had a brother, and so on, so for a little while I thought that once you had one kid you were locked in and they were all gonna be the same, and when I learned that that's not true I was shocked xD
On introduction of the fallacy at the beginning, I would say that there is one subtle but simple answer to it - Bayesian thinking
If I just say a roulette wheel spin Black 5 times in a row, that only happens on a fair wheel about 1 in 30-odd times. This gives a slight, but nonzero probability that there is something *wrong with the wheel*, that it is in fact not fair, but biased towards black somehow. The more consecutive Black that is spun, the more likely it is that there is some mechanical bias in the wheel. At some point, my confidence in this bias exceeds the offered odds of payout and it makes sense for me to bet Black exclusively, from an expected value standpoint.
The best part is, that if I am wrong, and the wheel is fair, then it actually doesn't matter which I bet on - if I was going to bet anyway on one or the other, then what I am actually betting on, in a sense, is my own confidence that the wheel is fair or not.
Essentially, it challenges the initial assumption that the wheel is fair. The gambler's fallacy accepts the 'fair wheel' as given and derives a wrong conclusion from this due to the way probability works - independent events do not influence each other. But the more information you gain, the more you can legitimately question your confidence in your initial assumptions - every result updates your confidence value in the fairness of the coin up or down, by some amount, even if it is small.
That is not the correct way to use Bayes. Bayes relies on a fundamental truth being realizable. Your example has no fundamental truth.
The problem with this thinking is the selection bias used. Why only consider the last 5 spins that were all black? To gather a truly valid view of whether the wheel was biased then all the spins of the wheel ever should be considered.
@@DumbledoreMcCracken never heard of this before. Could you explain or give an example?
@@MurdocK-BR read the wiki article on Bayes. It has an excellent example
@DumbledoreMcCracken
Neither the Wikipedia article on Bayes nor on his theorem nor Bayesian probability mentions fundamental truth. What example are you referring to?
Either way, Roulette wheel spins don't fundamentally differ from coin tosses. What makes you think otherwise?
My brother taught me a strategy he used playing lottery scratchers. He started by asking the cashier questions about the most recent (known) scratchers tickets sold on each ticket roll.
First, we find out if the last ticket sold on any roll was a winner, and they are automatically considered Red (No-Play) rolls. Next, we find out which rolls are priming to yield a winner by finding out which roll's last ticket(s) sold were losers, creating the Green (Play) list.
For the first round, we purchase one ticket from each roll on our Green list. After scratching each ticket, we take each winner and place their roll on the Red list.
In the second round, we purchase one ticket from each roll remaining on our Green list.
We repeat with subsequent rounds until each roll hits a winning ticket and no Green rolls remain.
This strategy should fall under the Gambler's Fallacy. However, we've maintained average net gains over time.
It would be absolutely _fascinating_ to see you evaluate and break down the math in our process.
That’s a sound system in my experience, which at the very least (given the bad overall odds of winning scratch off tickets to begin with) helps me play longer by not losing nearly as much. I do wonder if the gamblers fallacy applies to scratch off tickets as odd/bad as that sounds. Because each new ticket isn’t an independent event like the flip of a coin is. There a guaranteed outcome “known” in advance, prizes are distributed in some logical fashion; ie there aren’t 20 wins in 1 pack and 0 in another.
First of all - The gamblers' mistake was making any bet at all. Roulette is designed to take their money no matter how they play.
That said - If I saw a roulette wheel go black 16 times in a row and had to bet, I'd bet on black. 16 in a row might be a sign that the wheel is biased.
Exactly my thought. All the discussion in the video is conditioned on the assumption that the two outcomes are independent and equally probable. 16 blacks in a row would be enough to put the hypothesis into question.
That's an interesting statistical problem in itself - what is the probability that the wheel is biased given that the last 16 goes have all landed on black?
Or worse yet, that the wheel is rigged! Unscrupulous casino owner observes that everybody’s betting on red, and pushes the button that makes the wheel come out black.
@@vigilantcosmicpenguin8721 That's tricky to answer because it depends on the prior probability the wheel was unbiased. Bayes' theorem, IIRC, says that
P(A|B)=P(B|A)P(A)/P(B)
So, P(Wheeled is unbiased|16 blacks in a rows) = P(16 blacks in a row|wheel is unbiased)P(Wheel is unbiased)/P(16 blacks in a row)
It's _doubly tricky_ because
P(16 blacks in a row) is itself not independent of whether the wheel is unbiased. Typically the later problem is dealt with via the equation
P(B) = P(B|A)+P(B| not A), but this isn't really helpful in this case because it's just as tricky to assign a value to
P(16 blacks | wheel is biased), for the simple reason that "wheel is biased" encompasses a _lot_ of possibilities and we'd need to decide on a specific probability model to deal with them. If consider that we're dealing with a continuous probability distribution, and there's no reason we shouldn't, the probability that the wheel is _perfectly_ unbiased, i.e. that the probability of black is _exactly_ 50% is exactly 0, because the probability of _any_ individual value in a continuous probability distribution is 0 (if you aren't familiar with concept, 3blue1brown did a video on it a few years ago called something like "A probability of zero doesn't mean impossible"). In that case, what we'd really need to do is choose some specific margin of error and ask "What's the probability the wheel isn't biased by more than this amount, give 16 blacks in a row?" Or, in other words, "what's the probability that the true probability of black doesn't differ from 50% by more than this amount, given 16 blacks in a row?"
This is essentially unwinding all the nuance that classical hypothesis testing hides away in the silly and overly simplistic notion of "statistical significance" that essentially just uses
P(B|A) in place of P(A|B). I mean, the actual formal definition of statistical significance doesn't do that per se, but the concept itself really isn't much except to hide away the complexities that are simply inherent to trying to draw conclusions about P(A|B) based on P(B|A), which can be terribly misleading.
They wheel bias is controlled by the spinner person. So I'd have bet against everyone so I would win and the casino would get more money than it paid to me, but only once or twice least the spinner person make it go green
May I suggest, Jade, that you make a video on Monty-Hall problem. It took me good half an hour to figure it out, and it is amazing how many actual mathematicians fell into a trap of its "obviousness" when it was first presented. I would _love_ to see your take on it.
The problem with the monty hall problem is that people assume that monty hall is revealing a goat and offering to switch in order to persuade the player to switch after having selected the car, and believing that had the player selected a goat to start with, the offer would not have been made.
It's not like people are allowed to play 10 times in a row
If you run many trials of 10,000 tosses and then looked only at the ones starting with 10 heads in a row you will see that the percentage of heads tends to 5005/10000 = 50.05% not 50%. It does not trend back to 50% but the initial 10 heads just doesn't mean very much in a sample of 10,000.
Statistics textbooks also often mention that Kerrich also ran a test with a deliberately biased coin and the result converged to 70:30.
Does this mean that you can do better than 50% if you bet heads once after seeing 9 heads in a row and then no more bets until 9,990 more rounds have elapsed; then, also wait for another run of 9 and repeat? Why wouldn't this yield better than 50% odds? Would you have found a regularity embedded in the assumed randomness? 🤔
@@waylonbarrett3456 No. If the coin has a 50% chance of heads, it has a 50% chance of heads every time. There is no moment where betting on heads has higher odds than any other moment. That 50.05% that n graner talks about is the percentage including the 10 heads in a row. If you only counted the flips AFTER a streak you would still see 50%.
@@thomasfoster1985 I think you misunderstand what I am asking. This might be a clearer way to make the same point...
If 30 blacks in a row is considered rare and 31 blacks in a row is considered rarer, then does it follow that the 31st is more likely to be red? I know that if they are independent, then red is only 50% likely to be the 31st, but then why is 31 black in a row rarer than 30 black in a row?
You can't assert a regularity like one pattern being rarer than another and then act like the regularity is not there. The law of large numbers asserts a hierarchy of patterns in large sample sizes. It says 100 heads in a row should be rarer than 2 heads in a row, etc.
@@thomasfoster1985 I believe what n graner is showing is something real about rare events. The transition into and out of the runs seems peculiar and significant.
This also has a big role in sports! Although determining whether events are independent would be pretty challenging I imagine. Small sample sizes are also often used to make dubious judgments about how well a player performed
This is why a good player who succeeds at making low percentage plays no matter how entertaining it is, in the longer run turn out to be a bit disappointing once the luck runs out.
This gives rise to the "that player is on fire" fallacy.
@@JavSusLar I think that the hot hand is sometimes a fallacy and sometimes true.
Suppose a team is a on a losing streak, and they believe that a win “is due.” One might say this is falling in the trap of the fallacy, but interestingly believing in the fallacy might actually boost their next performance, increasing the likelihood of a win. I think this seeming paradox is resolved by seeing that - unlike flipping coins - players have memory of their previous games, and so each game is NOT an independent event.
I pondered this paradox when I was younger and realized that any outlier sub-sequence is forever embedded in the overall sequence, and its effect only vanishes after the sequence has so many samples that the outlier is too small to matter.
There is no guarantee of any counter-sub-sequence to exactly balance the previous outlier. The chance of such an outlier is the same as the chance of repeating the previous outlier!
Your content is always amazing and very easily accessible to those of us who are mathematically challenged. Thank you. Keep doing a great job my daughter is a fan.
Another way to say this is that after many trials, the statistical influence of a streak diminishes. The Cosmos does not need to revert the frequency to the expected one, it is that the expected frequency changes.
Think of it this way. After the first three coin flips are heads, the expected relative frequency for heads for four flips that began with that streak of 3 heads is 87.5%. The expected frequency for 6 trials that begin with a streak of three is 75% heads. Graph the expected frequency of X trials beginning with a streak of three, and when X = 3, the expected frequency is 100%. Then 87.5% for four, etc, until it approaches 50% as X approaches infniity. Graph the actual trials, and the actual frequency will roughly follow that line (curve) down just as it followed the horizontal line when no streak was baked in.
06:01 "contrary to popular belief, size does matter"
Finally a video where I don't feel like a dropout from elementary school ^^
Though I'd say the fallacy of those gamblers would've been to not bet on the winning colour. Either there is a reason why it spun to black so much like the wheel not being as balanced as everyone would hope or there is none and it doesn't matter what you choose (or there is something you don't know that just increases the odds for the opposite but didn't happen for random chance last time).
But *ACTUALLY* the real fallacy was to go into a casino, applying large sample rules and expecting to not loose all their money due to all the games being rigged against the player.
Yep those odds are rigged against the player. You'd be better off buying something you like than hoping to profit on terrible odds.
@@One.Zero.One101 yup. I believe only a few configurations for black jack and maybe poker against other players isn't completely rigged against you. But even then, if the casino staff sees you *actually* having figured out something to profit of the games they just throw you out.
Don't get me wrong though, I believe there is nothing wrong at going there with a set amount of cash, knowing to go out empty handed, have a fun time, experience the thrill of randomness with stakes and all.
I always love these videos from Jade. She is my favorite presenter and a real charmer plus super clear and engaging! 💖
Randomness works in two different ways depending on whether we care about order or not. The probability of "balanced" outcomes (half heads and half tails, for instance) being high is because we often ignore order, so flipping HHHTTT "feels" the same as HTHTHT even though it's a totally different specific order, and there are more possible patterns with 3 heads and 3 tails than, say, 1 head and 5 tails. But each *specific* order is equally probable (in an actually random system).
This is shown in Pascal's triangle, where the main numbers in the triangle represent the total number of ways to get to the same "macro-state" where there are the same number of each option. Playing with a quincunx/Galton-board will help folks understand how each different "choice" of going left or right fits into the larger possible set of patterns of specific "micro-states" that Pascal's triangle describes.
I really enjoyed this video. I realize how difficult it must be to constantly come up with new topics and to present them in such an easy to understand way. I salute you for always being up to the challenge.
Presenting them in an easy way is hard, but coming up with the topics is so easy. There are so many fascinating ideas out there in the world! I can't make enough videos for all th amazing ideas!
There's a mathematical/computer science concept called Kolmogorov Complexity that gives a sense of why 20 heads in a row "feels" rare, even though it's the same chance as any other sequence of coin flips. It's one of my favourite mathematical ideas. It's a bit weird to grasp though.
This probably goes beyond the scope of this video, if we're to stay focused on statistics and the gambler's fallacy...but thinking about the laws of large/small numbers, and finding these "clumps" of seemingly ordered results within a large sample of independently random events, which "balance out" over time...it strikes me how the law of large numbers seems to be, if not related, at least oddly similar to entropy. Though I have absolutely no idea if there is any connection at all.
yes they are very similar.
You can see how large number work on a averiges when you try to correct an averige, like averige gas milage by driving more efficient after a period of driving more ‘sportive’. The averige ‘solidifies’ with more trials, is less influenced.
At some point though, you have to wonder if it's more likely that the person who told you that it's a fair coin is lying. 10 heads in a row... plausible. 26 in a row... extremely rare. But after 100 heads in a row, in "real life", the odds are pretty good that the game is rigged.
The most I’ve seen is 13 reds in a row. Lost money betting black until 8 reds showed up. Switched to red after the eighth black and rode the streak figuring the game was slightly rigged. Sometimes you have to bet the hot hand.
There are no fair coins. However, a roulette wheel can have a recognizable bias and still lose you money because the all-lose positions overwhelm the bias.
I don't think it's possible to even make an unfair coin. If it's weighted unevenly, it still spins half and half - it's not going to speed up and slow down between revolutions. The bias in coin flipping i think comes from people who can barely get the thing to turn over when they toss it
Especially if there are cuts in the video between the tosses…
26 in a row is impossible for average person to witness. If you do witness it, the game was rigged
We once did the same experiment back in elementary school. A little more than 100 students each rolled a dice 100 times and recorded the results. After taking everything together we got an almost perfect result of 1/6 for each number. I loved this so much since it felt a little bit like magic
Thanks!
Maybe I'm biased.. but I think one of the most important subjects to teach universally is probability and statistics. It's relevant for virtually any inference you'd like to make.. It's nice to see this channel making these ideas mainstream. Excellent coverage of really unintuitive concepts.
I think formal logic would be good to include for the same reasons. It's crazy to me that logic is such a niche subject; I think it should be taught in some way from at least high school and beyond.
I completely agree. An intuitive understanding o multivariate statistics is very useful to understand society and biosciences. I would exchange it for some topics in geometry, trigonometry and logarithms.
Using formal logic would allow you to understand this video is hot garbage. I would honestly love to debate her or anyone on this topic. Flipping a coin 700 million times in a row with the hope it eventually lands on tails is not 1 in 2 probability. Your odds increase with each flip, and this is because the event occurred, not because of some pseudoscientific "memory" of the coin. I've also noticed people who call this a "gambler's fallacy" are most likely politically left wing like NDT, because certain aspects of their brain are either underdeveloped or have a mind numbingly obvious blind spot.
@@Ribcut ?
@@Ribcut your odds decrease with each flip though since your remaining chances for getting tails decrease...
Coincidently, just few days back I was reading "Thinking Fast And Slow" and now you have mentioned the same authors. Good work Jade !
Excellent explanation! The unanswered question for me, though, is when does a small sample set become a large one? If there is a middle ground, then how large is that?
When I was in school, the boys in my crowd liked to match pennies at lunch. My pals always lost to me, because I brought a hundred pennies every day, and they only had three or four each. My large sample size beat their small sample size every time, because I could afford to lose 99 times in a row, while they could only afford to lose four times. This is the real reason why the casino always wins.
Maybe it has some relation to her other video th-cam.com/video/zrFzSwHxiBQ/w-d-xo.html at around 5:45 min in where she explained how kinetic mechanics of a single particle can reconcile with the entropy of the whole system.
This is a great video. On thing I'd have enjoyed as an addendum to it, or a follow-up, is a discussion of the Monty Hall problem - which looks and feels like a gamblers fallacy question, but where the odds actually change with the choice.
An amazing video, Jade! It never ceases to impress me how our cognitive bias, prejudices and believes alter our perception of phenomena, though experimental evidence gives proof enough to question ourselves.
Many events in the natural world are not actually random but chaotic, consider water dripping from a faucet you don’t get one drop right on top of another that would be truly random. So in the natural world there is not independence between actions and the gambler heuristic applies. This would actually be a very interesting topic to do a video on.
well said. Bunches come in bunches. This is a big deal in real life.
Im honestly surprised how you can make such high quality videos in such short time. i love your videos keep uploading 😁
Yeah! Such a high quality video. She even included a "study" from our trusty Bill Gates!
@@humanrightsadvocate What was wrong with that?
@@humanrightsadvocate The study wasn't actually done by Bill Gates, it was done by the Gates foundation.
@@humanrightsadvocate You know sarcasm is invisible right? Bill didn't DO the study, he paid for it. He pays for a LOT of science research and you are weird.
i made experiment with my son. We counted in a row of flip-coins number of HTHH and compare to number of HHTH and HTTH we found them in different but sometimes overlapping rows. But number of them was almost equal. We made all categorizations that came to our minds. Then i asked him to write 50 results and separately note 2 arrays of real coin-flips and just by taking different type of measurement (for example number of H??TH - with two ignored flips in the middle) i showed him that even when we try then it is hard to do something randomly.
In 10,000 flips, having 200 more heads than tails is just a couple percentage points. In 1000 flips, having 200 more heads than tails is a massive swing.
My area is probability theory and finance, really nice to see this topic covered. A fun comment on the Levy arcsine law is “even the fairest of gambling games will seem unfair most of the time”. Indeed, when we price something like a forward contract in finance, it is expected to only be truly a fair bet at inception, all times after we would expect it to almost always be in one other parties favour. Path dependency is one of those common features that transcends all areas of applied mathematics (all areas of maths). I want to say thank you as my 9 year old asks me all the time what I do and this one (plus the zero knowledge proofs video) let me show him.
0:01 The reason Jaiden won
When I was in high school, sitting around class bored, I was flipping a coin, calling it in the air, and then scoring whether my call was correct or wrong. On one occasion that was never repeated, I called it correctly 32 times in a row, far exceeding the odds of your roulette wheel example, that's about 4 BILLION to ONE.
ESP
I've never understood why you would expect it too "balance out" if anything I would bet on black because the roulette wheel might be broken. The law of large numbers doesn't change anything, if you start at streak of 30 blacks and then do an other 1000000 spins you would expect that on average you see a total of 30 more blacks then reds. The absolute numbers don't balance out, the 30 just becomes negligible to the 1000000
What gets me is that if 30 blacks in a row is considered rare and 31 blacks in a row is considered rarer, then does it follow that the 31st is more likely to be red? I know that if they are independent, then red is only 50% likely to be the 31st, but then why is 31 black in a row rarer than 30 black in a row?
@@waylonbarrett3456 A different way to think about it is: Imagine we setup an incredible number of wheels and operators for those wheels (10^10^10^10 perhaps), and have them all spin 30 times. From here, we tell everyone who has not spun 30 black in a row to leave, congratulate everyone remaining on the incredibly rare thing they just did, and then ask those remaining to spin again. Approximately 50% of those remaining will spin into black again, the rest red. That is to say... 31 black results in a row is just as likely as 30 black results and 1 red result. One is *not* rarer than the other. :)
The sequence of 30 black spins in a row is kind of arbitrary. We care about it because it's simple in how recognizable it is. Really, it's not different to how likely you are to be upon the spin after the sequence "bbrrbrbbbbbbrrrrbbbrbbrrrbbrrb", or any other sequence. We only happened to recognize the first sequence because it fits a simple pattern, but that's just doing a disservice to all the other just-as-rare-but-not-simple-pattern-fitting sequences that came before it.
@@ReinOfCats OK, but I was thinking of this all in a frequentist/Bayesian way. Say I'm created 1 billion separate sequences of length 1 million. If I analyze each sequence, I find the probability of 30 consecutive black outcomes and also the probability of 31 consecutive black outcomes. Those numbers are different. Then, I do this with any 2 consecutive run lengths and I wonder why there is this structured hierarchy of probability if the wheel is statistically independent and is not influenced by this zoomed-out order.
This is better than other videos on gambler's fallacy because of how you went into detail about randomness being lumpy on small scales. Very helpful for conceptualising randomness. Thank you.
I have witnessed the gambler's fallacy a lot in the medical world, from both doctors and patients.
Anecdotally I've seen the fallacy mean that a patient and doctor have had two very different conversations.
Can cause quite a big problem when it comes to surgery and complication rates and such.
Tell us more. How does that conversation go? Is it the doctor or the patient who is more susceptible to the fallacy?
What are the odds I botch two surgeries in a row? 🤷♂️
Someone once told me they skip every tenth tablet because their doctor had told them that 1 in 10 people who take a tablet get a certain aide effect.
Anecdotally, a common lime of thought amongst doctors is 'ive never seen this adverse effect therefore it doesn't exist' which is in a similar realm of misapplying the laws of large/small numbers as the gambler's fallacy
@@robparker1742 Ah, indeed. If you only get 9/10 of the prescribed dose, you can expect to avoid the ⅒ chance of a side effect. Impeccable logic, that.
the real gambler's fallacy is stepping foot in a casino in the first place and placing a bet on something that has a 50/50 chance.
Most people calculate the probability of a 50/50 outcome sequence incorrectly. Let me explain.
In roulette, the probability of Red hitting 4 times in a row is 1/16.
However, the probability of four spins producing Red, Red, Red, Black, is also 1/16.
The sequence is easier to calculate the probability of when the outcomes are identical, and this is the core issue behind why most gamblers begin to feel as if they can predict the outcome of the next result. It is easy to miss the fact that any string of ten 50/50 outcomes is just as improbable as any other possible 10 iteration sequence.
This was a great video. Now I'm going to watch Arin Hanson feed his gambling addiction
I enjoyed this video. It reminded me of the time in college 50+ years ago at Lake Sumter Community College in Leesburg, Florida when one of our instructors (Pete Wilson) used a real life experience to demonstrate the law of large numbers. A group of us were carrying a cash box on the way to the gymnasium to sell tickets to some event. The cash box was dropped and coins went everywhere. Pete told us to match coins, one head with one tail, as we picked up the coins. The total should be very close to even... and sure enough it was!
This prompted me to do a google search on Pete. I knew that he had gone on to get his PhD and had taught at various top level universities around the country. Sadly I discovered that Pete passed away three years ago. He was a great teacher and I am sure is missed by his friends and family.
2:17 I think you mispronounced the word "adage". I'm not a native English speaker and also because English is not a phonetic language l look up words in an online dictionary and the dictionary pronunciation is different from yours. I don't know if this is because of dialect.
Her pronunciation was close to the French pronunciation (it's a word of French origin). Maybe that's the way they pronounce it in Australia? Or maybe she learned it from her partner who is French?
I’ve had this thought for years: that there seem to be some discrepancy between the law of large numbers and the independence. I’m so glad to see this!!
Birth patterns aren't random or independent events 🤔 pretty much skewed by biological factors and genetics.
I’ve been trying to find a resolution to this paradox for a while. I knew it was out there but couldn’t find it. Great video. Thank you
This is why I get frustrated with time travel shows and movies. If you could memorize the winning lotto numbers and go back in time, there is still the basic probability those numbers will occur. The laws of probability are independent of time. For example, if a person rolled a six sided dice and 2 came out and you went back in time and had the person roll again, there's still in 1 in 6 chance. There's no guarantee a 2 will be rolled. So if you were able to travel 5 years into the past, and then travel back to the future, you will arrive at a different future, since all probability that occurred between those five years would be "re-rolled."
It is worth noting that you described the coincident factors of events taking place as "odds," as in 50/50 or as a percentage and so on, while then noting that these odds do not always seem to stack up over longer sequences.
The sequential graphs showing an extended series of events coinciding are, if you think about it, nothing more than a visual representation of the process of arriving at an average of those extended events, where the mathematical "odds" of chance are confirmed by the longer pattern.
It would be interesting, for instance, to ask how many throws of a dice have to be made before the mathematical percentage of 6/1 is actually realised, and then to ask if this number of throws is always the same, or whether the same mathematical average can be identified from less or more throws on different occasions, and then to see if there are variables that can affect each the throws on each of these occasions.
For me, the point in question is not the odds of a single event, because I see this description as misleading, but how a sequence of single events is formed that gives the average, and how many variations on that sequence there are.
Heads and tails are said to be 50/50, but this is a guestimate that asks how many throws of a sequence have to be made to confirm that, and if there are more or less throws to the next confirmation of that same 50/50 statistic, does this highlight variables that are hidden by the pursuit of a 50/50 confirmation.
Fun fact: 97% of gamblers quit right before making it big!
1:13 - I did see a coin tossed by a soccer referee end up embedded in the turf edgewise. I guess that's the equivalent of "green" on the roulette... 🙂
There is a physical component to consider too, maybe the selection isn't random. Are fish with boils easier to catch? Is the black paint less polished than red?.
That is why taking notes in a casino isn't allowed.
"You've had three boys so the next one must be a girl." That's what my uncle thought before having his 2nd, 3rd, 4th, and 5th daughter lol. Or the church family I knew with *eight boys and no girls* in it - although to be fair the first 3 were divided between the parents' respective former marriages. Still for the mother 6 or 7 boys in a row is ridiculous lol
That must be how the Weasleys felt.
We stopped at four boys because at that point we couldn't afford another child!
Using ratio of sexes in a family is not really appropriate, because many or even most couples will have some kind of biologically created bias. The bias may be low, but it is not unusual for it to be quite high. So if you come on a family with 5 boys and the mother is pregnant, you really should bet on the next child being male (opposite to the gamblers fallacy) because it's quite likely that you're looking at some kind of systemic organic factor in the relationship between the medical or biochemical character of the two people.
I think things like epigenetics actually influence this!
I'd love to see a video from you on how the option to choose again between three doors is always better. That one always confused me.
First 😉
@@realitywave In Germany I was the first to click 👍 So, in a way I have already achieved this lofty goal. Maybe we can say: We were the first in our respective countries?
@@realitywave Next time you're going to be first again. 👍
@@realitywave I did it only for fun and hadn't even planed to be the first. I just like Jade's very informative and well researched videos.
@@realitywave As I am a teacher in adult education myself I can only back that up 👍
When I was at university, in an initial physics/physical science course, our professor had been running an experiment in the course for over 20 years. Every class, every day, every semester, students, in lab groups of 4, dropped 100 wooden match sticks, in groups of 10, from a specific, standard height, toward a broad red line in the center of an oblong, metal baking pan. Of course, all the pans, liners, red lines, matches, height, blah blah blah, were identical. The test was to count how many matches from each drop touched the red line. Each lab group tracked their data & reported it to the professor each week. He, in turn, tracked all the data from all the groups for over 20 years, and his graph looked a lot like yours! He demonstrated that, as time progressed, the instances of red line contact approached average - while our individual lab samples were often not average at all.
To me, one of the best things of Breaking Bad was that, having lived in Albuquerque (although I left before the series started), I know those places. I used to go to that car wash. Saul Goodman's office, at least the external set for it, was right next to a stripper joint, which is such a Saul Goodman thing.
This reminds me why my statistics teacher was so touchy with semantics. A change in wording changes what you are talking about.
I think you are one of the best STEM educators on TH-cam Jade. Thanks for all the great content.
Another way to think about it is in terms of macrostates/microstates. Describing the situation of say, "8 heads and 2 tails" when flipping a fair coin ten times is a macrostate, but a particular ordering of that macrostate like "4 heads in a row, followed by 1 tails, followed by 4 heads, followed by one tails" is a microstate; the macrostate description has more microstates (because you can switch up the order) than a macrostate of "10 heads", which is why there is a lower probability of flipping 10 heads in a row, than there is of flipping 8 heads and 2 tails, but any particular microstate has the same probability as flipping 10 heads in a row. The reason why macrostates with more numerically balanced distributions are more probable is because there are a larger number of possible arrangements which suit that description of the overall outcome.
So, the reason why you aren't more likely to flip tails after the fair coin lands on heads 9 times in a row is because you are then no longer referring to the probability of flipping 9 heads and 1 tails, but instead, the probability in question is that of 9 heads in a row followed by 1 tails, which is 2^(-10), identically to the probability of getting 10 heads in a row.
1:45 The chances of getting 3 heads in a row is 12,50% while the chances of getting 4 heads in a row is only 6,25%. Wouldn't it be safe to assume that it stops showing head since the likelyhood of this scenario happening is unlikely?
For example: the scenario of 10 heads in a row occuring ist around 0.0977%. This scenario is very unlikely to occur, so if I had 9x head in a row I would definitely assume the next one will not be head anymore since I would not think that this 0.0977% is happening right now. I know that in a way it's still just 50/50 and the previous events shouldn't affect the newer ones since they are unrelated in a way... but at the same time I also think there is another factor that noone is thinking about when it comes to this statistics game.
For schools, you can also get strong selection biases. If you look at the SAT scores for schools in different states some states with pretty bad education systems have pretty high average SAT scores. When you look closer at the data though, states with bad education systems don't encourage, or even actively discourage poor students to take the SATs, while other states encourage all students to take them. This brings down those states averages, even if they have better schools overall.
A good way to think of the balance between the law of averages and the law of big numbers... say you start flipping a fair coin, and by dumb luck you start with a string of 10 heads, but then suddenly some sort of temporal event causes your timeline to fracture into a million timelines, and you continue, in each of those timelines to flip the coin until you have 10,000 flips in each timeline. On average, at the end of all of those flips, your totals will have 10 more heads than tails, because, on average, each of those timelines will be ruled by the law of averages- but, the law of big numbers will mean that your small fluctuation of +10 heads is a very small portion of the total flips, while the law of averages will mean that some of the timelines will be +- or = to perfect 5k to 5k splits.
Of course, the smart thing to do if the wheel keep landing on black (and you are going to be dumb and gamble in the first place) is to watch for any tells like the table manager flipping a switch, and, in the absence of that, maybe give black a couple shots. There is always a chance the table is rigged or no longer fair for some reason, in which case you might as well get in on it (well, depending on your feelings about ripping off a casino).
Strictly speaking, the law of large numbers only applies as the sample size approaches infinity. Also, the word „relative“ does a whole lot of heavy lifting in „relative frequency“. For instance, if you integrate over accelerometer values in order to find out your velocity, the noise adds up rather than averaging out. You‘d need to average the values, but then you don’t get the velocity anymore.
In the opening clip, it might be the case that the roulette table was biased. Apparently, through repeated wear and years of use, the ball sometimes prefers to land on some spaces compared to others. Maybe falling for the gamblers fallacy of "It fell on black so it will continue falling on black" and always betting on black would be profitable assuming that the probability of landing on black is greater than 1/2 (due to the wear).
Which is why scientist have to "publish" your findings... Does multiple things... One of which is "showing the data" which allows others to review how samples were taken and the scope of the data collection that support or refute those finds... In other words, testing how good the tests used were!
I have never been disappointed in a single one of your videos! Between that glowing optimistic smile and the solid, well crafted subject matters covered by very clear and concise breakdowns - making each topic manageable and relatable!
Thank you.
Thanks! This is one for my toolbox 'I'm I wrong?'.
Filled tools for countering my own biases ;)
Thank you so much for this video. When I heard of Gambler's Fallacy and of the fact that as you approach infinite trials the result approaches the probability I was highly confused. It seemed contradictory at the time.
It's been about 7 years since we talked about the law of large numbers in school and this has always been something that I was wondering about ever since.
The movie Jerry and Marge go large talks about this. If you find a flaw in any kind of betting you can’t take advantage of that with a small number of bets. You have to place enough bets for the law of large numbers to kick in.
Great overview. It helped me think of it more deeply.
My experience with a gambler , who's attitude seemed to be : something will happen or it won't happen, so on every spin on the slot a jackpot will "hit" or it won't "hit". No math required.
I suppose calling a fallacy on any attitude or system a gambler has is contingent on a winning or losing outcome. (me being subjective).
Your presentation on "Representativeness Heuristics" reminded me of a Star Trek episode , "The Arena". Kirk and a Gorn are forced to battle each other by a higher entity to settle their dispute. Kirk finds (his initial Gorn "profile") the Gorn, with it's reptilian features grotesque and incorrigible (heuristics?) . Although Kirk triumphs in battle over the Gorn , he decides not to kill it and asks the higher entity not to destroy it either. The higher entity is impressed with Kirk for showing the "advanced concept" of mercy, and express's "there still might be hope for the human race".
People seem to forget that a roulette wheel also has a green space, which is what gives the casino their edge.
My uncle Whizzo was a sucker for the gambler's fallacy. He lost a pile of money wagering on professional tic-tac-toe, he was always convinced that somebody's gotta win eventually.
I like Feynman’s quote very much about this.
I had this demonstrated to me in school with coin flips and noting the results down, and she said if we were here all day the numbers would probably get close to 50/50, but we can’t say when any of those balancing moments would occur and it might oscillate or might look one-sided all the way (as born out by your simulations).
I thought that did a pretty good job explaining how a thing can be evenly fair overall but randomness is more about when in the overall sequence we witness. Are we in a run of heads time or a run of tails time? No way to find out except to sample it.
A couple of thoughts:
Firstly, while the relative frequencies do tend to converge on 50%, the absolute frequencies tend to diverge. After ten thousand coin tosses, you'd expect either heads or tails to be about 100 ahead of the other. For a fair coin, the expected lead is the square root of the number of tosses, so the "error" in the relative value is roughly the square root divided by the number of tosses, or one over the square root (well, half that because if heads is ahead, half the lead is excess heads; half is a shortage of tails).
Secondly, one possible contributor to the gambler's fallacy is that there is a similar scenario where red and black are guaranteed to even out in absolute terms, not just relative - and that's drawing (without replacement) from a (well-shuffled) pack of cards. Each individual card drawn is equally likely to be red or black, but drawing a black card makes the next card more likely to be red, and vice versa because you have to end up with twenty six of each colour when you reach the end of the deck...
Yes but that is not an example of statistical independence
@@UdoLattek02 Yes, that's why it behaves differently, but for someone just thinking casually about how it would turn out, the two look pretty similar (particularly if you only look at one or two draws/spins)
Brains use bayesian inference which is a good way to think about casually connected events.
It's misapplication with independent events is the basis of the gamblers fallacy.
We've watched a "large number" of your videos. Seems the "uniformity" and "balancing out" is that this is a fun channel. Thanks. Cheers.
This reminded me of when my coworker told me how they pick lottery numbers.
They look at past results for the numbers that have come up most often in the past. And then specifically picks those ones because they're clearly more likely to come up.
But then, on the flip side. They also make sure their specific group of 6 numbers has never won before because, "The same set of numbers will never come up twice".
I found this strange because not only are we assuming the rolls have memory, we're assuming independently, they're weighted, but in sets, they're due.
Obviously both strategies are incorrect and don't help you win. But since the odds of any set of numbers is the same anyway, at least their strategy is fun to figure out and doesn't actively hurt you to do. Unrelated to the video, they also refuse to take numbers in sequence. And always buy two tickets only 1 number off because, "you always get a ticket and miss by like 1 number!"
so with a roulette wheel there is a bunch of factors that could lower randomness, friction, wheels levelness, the human factor, and if you see a pattern emerge there could be a reason for it
Yaaaay! I really missed your little cartoons. Glad to see them again ❤️
Say in the first 10 flips you got 9 heads and 1 tail so your frequency of heads is 90%. But then in the next 2N flips (where N is large) you expect roughly N heads and N tails so your frequency of heads becomes (N+9)/(2N+10). If you graph this function you'll see it starts at 90% but gets closer and closer to 50% as N increases. So the coin doesn't have to be biased in favor of tails after the first 10 flips for the observed frequency to revert to the expected frequency. (See also: regression to the mean)
Yup, although it may be more accurate to say "regression towards the mean"
An interesting corollary is: given a run of tails, say, with what likelihood can we reject the null hypothesis that the coin is fair? If you flip thirty tails in a row in fifty trials, you cannot blithely invoke the gambler's fallacy and say the next flip is 50/50, because it is reasonable to ask the probability of observing that outcome by chance. If the likelihood is too low (for a given value of low) we reject the null hypothesis that the coin is fair. This is when Bayesian statistics enters the picture. Note this is different from asking what the likelihood is of flipping 30 tails in fifty trials - that's not really all that unlikely.
8:16 you said “it was still the wrong move” but actually there was no “right” move. It’s always 50/50.
For further reading I recommend Nassim Taleb's Fooled by randomness.
I just rememberd something really fitting: At some point a Phone-Company that i do not like, implemented a random shuffle system in their Music service, but many many Consumers outraged at them telling them the shuffel is not random enough because many songs got playd multiple times at a row. So in the end that "Phone company i do not like" decided to make their already random shuffel less random, to please the crowd into thinking its more random.
After a really very strong showing in the video's contents, that jingle to play us out was a canorous delight indeed.
There is a difference between chance and probability. Chance in that instance is 50/50, but probability as bounded by sequence length. We tend to look at things in small snapshots of time rather than over longer sequences of time and larger numbers of trials.
I don't know if you've upgraded your lighting setup or I've not been paying as much attention in your previous videos, but this video is ✨ wonderfully lit ✨
I think a nice visual would be showing the same experiment (graph of coin flips/dice rolls) that already starts with 100 heads or 100 6’s.
It would also converge to 50% and 1/6 respectively, even with the initial “lucky” streak.
damn you should speak to my animator
I often wonder why I was forced to take statistics to get a college degree when in my jobs managers have ignored anything in which I used statistics. I once showed that based on failures in previous products, our lack of testing indicated that we would have a more than fifty percent of having a major bug in a product we were about to release. The managers ignored me and released it but we got lucky that time. My boss send out an email saying that since there were no major bugs, my "silly mathematics" were "wrong".
"Randomness is clumpy." Great quote!
Have been thinking about this recently. Thanks for the clarification!
I've heard this explained multiple times but you did the best job and so I subscribed.
A coin flip is actually slightly off from 50/50. The side facing up has a slightly higher chance of being up again when it lands.
You can't assume that everything that's supposed to be random actually is. Roulette wheels can be fixed, if the bettors are making lopsided bets on red or black then a dishonest croupier can simply select the other color. In the case of a casino where there was an excessively long streak of black while the majority of the bettors picked red the smart thing to do would be to pick black. If the wheel is honest then your color choice doesn't matter, you'll win just under 50% of the time if you stay long enough, the house edge is the 0 in Monte Carlo, and 0 and 00 in Las Vegas. However if the wheel is fixed then you should bet against the crowd. If everyone else is picking red then bet on black. If the crowd switches to black then switch to red.
The thought that occurred to me watching this video was that the entire field of Statistics is based on the Gambler's Fallacy in reverse. The basic premise is that you take a representative sample and then based on the relative sizes of the sample vs the total population work out how likely it is that the sample represents the mean of the total population. There is a little more to it than that, but that is the basics of it. Which just reinforces the cognitive disonance of the Gambler's Fallacy when one doesn't stop to acknowledge the fact the events are independent random events.
Always bet on black - when playing roulette😁👍
Awesome video Jade. Thank you😁