Alright hopefully this video helps put this all together! As mentioned by wikipedia it was very hard to see why my initial assumptions were wrong. But thank you to everyone in the last video for the discussion about what is really going on within this problem. Also I don't really talk about the 'born on a Tuesday' part in this video but the same math I talked about would apply (asking would yield a change, whereas someone just mentioning it would keep the probability at 50%).
The original confusion comes from not stating explicitly the question that brings about the information in each cases. This makes calculating the probability of the given state confusing. The case where the dad tells us "randomly" about their daughter is the result of asking: "tell me about one of your children". He could have told us about a boy. The other case is the result of the question: "do you have a daughter?" He could not have told us about a boy. Once these starting questions are laid out, the probabilities are easy to calculate. Same thing with the name: - "What is the name of your daughter?" He could tell us the name of the other daughter if he has one. - "Do you have a daughter named Julie?" He could not have told us about the other daughter. And same thing with the day of birth.
There's 99.99 % chance they are gonna kick your ass and 0.01 % chance that they love money more than their daughters. I still don't think 10 dollars is enough.
Just don't go into a bars full of statisticians when you make these bets. Only the ones with two daughters will take you up on the bet and you'll be wondering why you're having such a bad run of luck 😂
What is the game theory optimal (GTO) behavior of full-knowledge statisticians playing this game -- on both the side of the interrogator/bettor and the bet-acceptors?
The way I made sense of it in my head is this: When he's talking about his "2 kids, one of whom is a girl", the "one" in question could be one of 2 people if he has two girls. As soon as he says the girl's name, or sees her and says "there she is", he's definitely talking about one specific girl. So the shift comes from a subtle change in meaning of the word "one": It changes from "either of my daughters if I have two daughters, or my only daughter if I have one of each" to "this specific daughter". To me, this explains the "paradox", and I guess it kind of corresponds nicely to this video's explanation that half the time he'd say "Julie", and half the time he'd mention the "other daughter". Great video though, it really helped me to start thinking about the paradox in the right way, since I've been aware of the puzzle for a long time and never really felt I'd made proper sense of it.
I came across the prev video today and discussed about the paradox with my friend. We came to a similar conclusion ourselves, but slightly different. When the name is not told, you are guessing whether the parent has another daughter. But once the parent points/name/specify Julie, and ask whether the other child is a girl, we may think that the question is not changed much but the it is indeed changed. The question now becomes what is the chance of having another daughter who is not Julie?
@@stonecoldxi9138 If family has 2 children and we see one of those children, a girl, then the chances of other child being girl is 1/3. But if we know some additional data about that girl, for example that it's an older or smarter or better behaved child, then chances are 1/2. Assuming boys and girls are equally smart and well behaved. But name is not important. If you know girl's name, chances are still 1/3. Unless there is a law that a family of 2 children must name one daughter Julie.
Psywriter That probability is possible to calculate but a lot of deep research and looking at the census and such to find that probability. Unless we assume there's an equal probability of being born in any state
Well done! You nailed it. There is so much shallow misinformation about this and related problems. Might be typical for the internet, but strangely enough even some math professors got this one plain wrong. (as was the case for the Monty Hall) I was going to write you to clarify, but writing is hard and now you already did it in a very nice way. Note 1: As a physicist, I like to think about it this way: "asking people to step forward" is a preparation of a "state" or "ensemble" i.e. a set of possibilities "asking (specific) questions" is a measurement on such a set. If a problem is given in natural language, we always have to clarify which part is a preparation and which is a measurement! Note 2: I only noticed one misspeak at 10:12 in this video. You should have said: "If you --outright-- ask if they have a daughter named X, and you guessed correct, you get 50%" If you first ask for "having a daughter" and later guess the name, the 33% still stand. So, again no "jump" in probabilities!
Thank you! Yeah the monty hall problem took me a few hours to fully grasp but this one took me a few days cause the details (in my opinion) were even more subtle. Really interesting problem though and glad I came across it.
@@zachstar Monty Hall is my biggest source of anger. Monty opens a door and shows you a goat. **You then say, "Hey, Monty, do you always open a door that has a goat behind it?" and Monty says "Yes, I like messing with people."** Now you switch. Without knowing that Monty always does that, there isn't enough info.
@@zachstar Indeed. Every analysis always assumes that because Monty _can_ open a door with a goat, he always does, thus the usual percentage. But the problem as usually stated never says so. Sigh. Great video, BTW, pointing out the difference. Human language is so... tricky.
@@AviDrissman Why would he open any other door? He won't open the one with the prize, and he won't open yours, there's literally no other door for him to open that isn't a goat.
Outstanding! I knew the comments on your first video would be heated.. But this video explains the subtle details better than any I've seen. Well done!
Actually if the sentence they utter is "I have a daughter," there is roughly a 0% chance they have two daughters, because if they had two daughters they would say "I have two daughters." That's how actual people talk.
Daniel Copeland nah, you can say something like “my daughter just graduated high school” which doesn’t rule out you having another daughter and someone can say it
@@tcoren1 "My daughter" and "I have a daughter" are not the same utterance. Only one of them is a complete statement, for one thing. Yes, the phrase "my daughter" (unlike "I have a daughter") doesn't put an upper constraint on the number of daughters the speaker has. But notice that your sentence does not convey the information that the speaker has two children. (If they said "I have two kids; my daughter just graduated high school," the natural way to parse that sentence is that the person has exactly one daughter, because otherwise they wouldn't have used the phrase "my daughter" to single out one of the two children. They would have used some other expression that did distinguish the two.) Face it, this just isn't how people talk. Natural language abhors this particular kind of ambiguity.
Daniel Copeland nobody who has two kids would ever say “I have a daughter” in natural conversation, but they might say “my daughter” it’s true that this doesn’t include the having of two children, that should come from the rest of the conversation
Thank you for making these two videos. I saw the first one before my time in the back of an Uber today, and I admit YOUR first part popped into my mind, and I was so happy to see the second part when I got home today. Critical and detailed thinking skills, well, our civilization needs all the practice we can get! And a friendly style with some humor is one of the least threatening ways to go about it. Rock on!!! Oh, and hello from the Bay Area!
The way it dissolves for me is if the person says “one of my children is a girl” vs “my first child is a girl” (or of course, “my second child is a girl” which is the same). The pool of possibilities are GB,BG,GG for the first, and GB,GG for the second. I think all the scenarios are subtle variations on this.
Simple foolproof test: Does the fashion in which you got the information make a difference between boys and girls? If the info is delivered randomly, no. So it must be 50%. If you asked specifically for ‘girl’, yes. The prob can be different.
If 'one of which' is a daughter, than there is a 100% chance that the other one is a son, because of otherwise, they would say two of which are daughters
Ofcourse it doesn't come up like this in conversation but someome could mention they have two kids and then later say something like: I saw my daughter today.
In natural language I think I agree with you. I strict mathematical language I'd disagree. It is like the word "or". In natural language, it is one or the other, but not both. In mathematics, it is one, the other, or both.
I think this is weird: I was just explaining this to my better half and she _just. does. not. get. it,_ BUT not for the reasons that you'd think. The usual confusion over this one is the same as for the Month Hall problem: "logically"we conclude the answer to be 50:50 and 33:67 seems counterintuitive - but not my wife... She starts with: "But you are assuming that boys and girls happen equally often, and that if you already have one girl that the next is equally likely to be a boy or a girl." I explained how spermatogenesis gives almost perfectly equal split, but even if this wasn't the case, for the purposes of the example it isn't important - this is a "perfect" hypothetical - assume 50:50. Then I talk about the "Julie" issue: "But what if her name is Karen?" I explained that it doesn't matter, the example works regardless. "But what about Esme? That name is very uncommon these days." Still doesn't matter, your just need more families to capture enough 'Esmes'. "Oooh, but that is different - you didn't say anything about getting more families." So I explain again (and again and again) about this being an idealized scenario. As such, where it is not stated, the assumption is that any confounding variables are perfectly average. i.e. the two groups of families (BG and GG) are indistinguishable, that having one girl doesn't change the way the parents would choose a new daughter's name - independent. "But you didn't say that!" I have degrees in physics and engineering. My wife is a PhD academic who uses *_ QUALITATIVE_* methods in her research. Apparently basic "quant" doesn't compute to a "qual" researcher. Exactly ZERO of the normal assumptions I make (and I assume most others here too) are assumptions that she makes. Bizarre!!
funny stuff. but yeah, I agree agree that this girl girl problem has the same common sense paradox as the Monty Hall problem. If a drunk runs on stage and opens a "random" door and Monty announces "wild stuff! I was about to open a door to help you but now I'll skip that!" you just lost your 2 to 1 advantage in switching doors, you get 50/50 either way.
lol she's really getting hung up on the name. The name is not actually what matters. It's the act of specifying a girl at all at the beginning that changes the probability. You could say, "I have two daughter, one of which is...standing right there." This changes the probability in the same way because you are specifying a daughter that is 100% exists
The answer is, again, as I believe, not 33.3% because we know that all of humanity is unique and cannot selectively pick and choose. Girl A and Girl B can be interchanged to find a 50% probability in your original problem, and labeling them is necessary because they are unique and it correctly identifies the probable situation.
yes, but "one is a girl" and "at least one of which is a girl" are different, though one could argue that "one is X" is ambiguous and could mean both "one (and exactly one) is X" and "one is X (but I leave the other one totally open)"
I like this follow up. The way I like to think about it is whether you’ve received new information. Asking gets you new information, whereas when you just let people talk, they’ll tell you things randomly in a way that reflects underlying probabilities and you won’t get new information
What do you mean, "you won't get new information"? You do get new information, just not information that may be useful in Zach's particular trick of scamming people out of money.
@@limonick Nah, I think his understanding (as is the case with most people's understanding) of "new information" is just faulty. When you approach a guy at the bar whom you know has two kids, and the guy hasn't spoken yet, then the odds of him having two daughters versus him having only one daughter are 1:2 ; he's twice as likely to have a daughter and a son, compared to having two daughters. When you ask "Do you have at least one daughter?" and he answers "Yes", then you do receive a bit of new information because now you know for certain that he doesn't have two sons, but the odds between "two daughters" and "one daughter and one son" haven't changed; they remain 1:2 .So in that sense, you actually _didn't_ receive "new information" regarding the odds between "two daughters" and "one daughter and one son". Whereas if the guy spontaneously says "I have at least one daughter" (or "I have a daughter who likes [insert some hobby here]", or something alike), then the odds do change, from 1:2 to 1:1 (or 50:50); so then you did receive "new information" that changed the odds. Only this type of information is less suited for the particular exploitation scam (money bet) that this video creator was thinking of. It's similar to the case of the Monty Hall Problem. In that problem, it makes a difference if Marty (the game show host) knows what's behind the doors and always reveals a goat, or if he doesn't know himself what's behind the doors and the fact that he revealed the goat was just coincidence (because there was also the possibility that he would have revealed the car). In the first case, it's attractive for the contestant to switch to the other remaining door because that door is twice as likely to be the door with the car, as the door of the contestant's first pick. In the second case, it doesn't matter if the contestant switches or stays, because both the other door and the door of the contestant's first pick have a 50% likelihood to be the door with the car. Some people then word this concept as "If the host doesn't know what's behind the doors and he happens to reveal the goat by mere chance, then you didn't receive new information. It's only when the host does know what's behind the doors and always opens a door with a goat that you receive new information, which changes the odds." Well, no. In fact, it's the exact opposite. If a knowing host revealed the goat by design, then the probability that the door of the contestant's first pick is the door with the car hasn't changed: it was 1/3 (or 33.3333...%), and it remained 1/3 . Whereas if an unknowing host revealed the goat by mere coincidence, then the probability does change: from 1/3 to 1/2 (= 50%); in other words, you/the contestant did receive "new information" that improved the chance of the door of the contestant's first pick significantly. The key thing is: "new information" doesn't always mean "gaining an advantage". New information may actually increase _disadvantage_ . (For example, when Marty knowingly opens the door with the car and says "You lose", then you are receiving new information that changes the probablity of your first picked door from 1/3 to 0 .) However, many people, including (in my view) the commenter above, have trouble making that distinction.
Another example of this comes from the card game bridge, where you are dealt a hand of 13 cards. If at least one of your cards is an ace, the probability you have at least two aces is slightly less than 1/2. If at least one of your cards is the ace of clubs, the probability you have at least two aces is slightly greater than 1/2.
This probability stuff is really hard to get ones head around, so it's nice to see a thorough explanation. I think some of the Markov chain examples might benefit from an additional representation as a probability tree to add another perspective, if you decide to make more videos on such subjects.
One note about that betting game: Who's more likely to accept your bet? - The strangers who are not sure why you're asking them about their daughters and making bets about them? - Or the stranger who thinks they already know what your angle is?
Thank you. After being presented with the paradox of how adding days to the the mix effects the probability I spent a lot of time thinking about it and thought that probability must be broken. This video made it quite clear to me.
It's actually easy! Think of it as flipping two coins instead . If I say that at least one of my coins came up heads there are three possible states: heads/heads, heads/tails, tails/heads. BUT if I also say that only a single coin which came heads has an X marking it, then there are FOUR possible states: xheads/heads, heads/xheads, xheads/tails, tails/xheads. By telling you that ONLY one of the coins that came up heads has an identifying attribute, we've made it possible to differentiate between both head/head combinations, which changes the number of identifiable states, and thus the probability. The reason this seems so counterintuitive with the human example is that we "assume" that we can tell two people apart, but the statistics don't make that assumption which is why the probability changes once one of the children becomes identifiable and we move from three possible states to four: Julie/girl, girl/Julie, Julie/boy, boy/Julie.
@@TheTim466 I'm just using the pair of coins to demonstrate each valid state that they might be in, given specific information that we have. We can assume that the chance of any single coin alone being heads is 50/50. If all we know is that at least one coin is heads then there are three possible states and only one involves the second coin being heads so the chances are 1/3. If we additionally know that the coin which is heads has an X on it (the coin has been "named" just like our Julie example) then there are four possible states for the two coin pair to be in, and two of them involve both coins being heads so the chances of coin two being heads is 50/50
If a proof is a reasoned argument that helps people to accept something, then this seems to be a good example of how human psychology and wording can make a difference to the quality of the proof.
it's the difference between saying "I have 2 daughters and I'm randomly thinking of one of them" vs "I have 2 daughters but I'm being directed to think about a specific one" And similarly "I have a boy and a girl and am randomly thinking about one of them" vs "I have a boy and a girl, and am being directed to think about the daughter"
I guess the best explanation is to look at who comes up with the data: If you pick the gender, you're performing a postselection, which can introduce correlations to the dataset (you're excluding 50% of cases with matching genders, so the remaining cases with matching genders are fewer in total!). On the other hand, learning data from all samples (gender-of-random) cannot change independent data in the same samples (gender-of-other). Now, the Julia case is interesting in that if you're postselecting on a property that has zero chance of occurring (each of infinitely many names is equally likely), you end up with no samples and you can't do probabilites over that, so you have to take the limiting case (each of finitely many names is equally likely, but what if we keep adding names). That you can't introduce correlations if you remove almost no samples is straightforward. That you can't introduce correlations in what's left if you remove almost all samples in this specific way is somewhat interesting, but I do suspect that once you define "this specific way" rigorously enough, the non-correlation falls out naturally (as it does in the next paragraph). But the cleanest way to retrieve our conditional probabilities is to use geometry: If we sort the names by gender first and then by alphabet, we can place all samples in a square. Then {first-is-girl} is a rectangle with two equal parts, {GB} and {GG}. {any-is-girl} is a triomino composed of {GB}, {BG} and {GG}, so {GG | any-G} is 33%. {first-is-julia} now becomes a line in that square, and {any-is julia} becomes a union of two lines. And each of these lines is split evenly by the boy/girl line.
I feel like the simplest explanation of this is that the verbal informing of the name isn't what changes the probability. It's the fact that you're restricting the set of available parents you _could_ be talking to to ones that have daughters named Julie, and _that_ is what makes the probability 50%. In the general population, the odds are 2:1 (for the reasons early in the first video... G:G, G:B, B:G). In the subset of people with a daughter named Julie (or any other same name across the set being considered), the odds are 1:1 (again, for the reasons explained in your first video.) The explanations for each probability are spot on. It's the attribution of the change to the "telling" that causes the apparent 'paradox'. It's a super interesting problem to think about from the perspective of... well... perspective. But from a probabilistic standpoint, it's pretty easily explained in the first few minutes of the original video if you take the focus off of the "telling" and put it on the selection process in the first place.
(For example, you can write a pretty simple program that demonstrates the general 2:1 ratio with randomized data. When you add an additional conditional that checks if they have a daughter named Julie before even prompting a guess, the results will become a roughly even split of 1:1.)
Here's one assumption to make, which I've seen in some versions of the problem and wasn't discussed in either MajorPrep video: What if you assume the father says, "I have two children, at least one of whom is a (boy or girl)", regardless of what type of kids he has? Obviously, if he has two boys, he'll say "boy", and if he has two girls, he'll say "girl". Further assume that, if he has one of each, he'll flip a coin to decide if he'll say "boy" or "girl" (50-50 chance). In this case, if the father says "...at least one of whom is a girl", the chance that both children are girls will be 50%.
An easy explanation to this is if information is presented about one child, it depends on how it was presented and by who and with what knowledge and intention. If you hear the name of one of the children who happens to be a girl, the odds of two daughters is now 1/2. However, if someone intentionally gave you the name of a daughter to keep you guessing the number of daughters, it’s still 1/3. It’s the same information but presented in different ways and therefore has a different effect on the odds. It’s like the Monty Hall problem had the one door revealed by the host who has knowledge of the correct door and a purpose of concealing it. Hence the chances of getting the right door by switching was 2/3. However, if that same door were revealed by a random audience member with no knowledge of what’s behind each door, the chances of getting the right door by switching would be 1/2. Again, same information, but different circumstances. Hence, different odds
We should notice that between the two cases of "let US filter for properties we are interested in" and "someone has come to us declaring a property we are NOW interested in" there is already a pre-selection going on, and if that person has to fit multiple properties, the more they fit them, the larger the chance they come to us with that declaration in the first place. In the latter case, the other side decided the metrics we are interested in, which they by definition fit. If a WW1+WW2 veteran who is 120 years old now, was an astronaut, knows 20 instruments and 40 languages comes to me with all that information, the chance of all that occurring is 100% AFTER THE FACT, but if we are interested from the getgo whether someone like that even exists, well, good luck with that. That's the level of distortion that can be caused in the probabilities depending on what the initial group we look at is.
Ok. I think I've got it in an intuitive way: it's really the difference between the odds of a pair and the odds of a particular child. Siblings genders are (we'll presume) independent, which is how we get the 50/50. But when you're talking about a specific sibling, that swaps the perspective and it really does become like the Monty Hall problem.
If I ask a guy in the bar if he has a daughter named Julie (and he does), he's not going to say yes. He's going to start a fight with me, or walk away, or get really scared. He's not going to go "I actually do. You want to guess the gender of my other child?"
Going back to the situation in the first video, where the guy says "I have a daughter called Julie." You have 10,000 families and the chance of a girl being called Julie is 1 in 100. Of the 5000 BG/GB families, 50 will have a girl called Julie. Of the 2500 families with two girls, 25 will have a first-born called Julie. These parents will not call their second daughter Julie. So there remain 2475 families who may call their second girl Julie, so there will be 24.75 second daughters called Julie. To get even numbers, start with 1,000,000 families, with 500,000 BG/GB families and 250,000 GG families. From the BG/GB families we have 5,000 Julies. From the first-born of the GG families we have 2,500 Julies. But the families who have already named their first daughter Julie will not so name their second. That leaves 247,500 families, of which 2,475 name their daughter Julie. So the chance that a family with two children, one of whom is named Julie, will have 2 daughters is 2475/2500+2475, or .4987 (if my arithmetic is correct.)
You are exactly right. This was my main issue from the original video - the lack of a perfect probability. I had to assume that Zach was being lazy and just rounded to %50. Excellent job sir!
Thanks for an interesting follow up. Following our discussion on the previous video, I was motivated to write a Python script where I simulated some scenarios. I created a large number of "Mother" objects, each of which has a random number of children (I weighted each of these according to actual statistics on how many children American women tend to have each, though of course there was some rounding off for simplification) and each child is assigned a random day of the week as their birth day. I then used 200 different names ("Name1" through "Name200", where odd names are assigned to males and even names are assigned to females). It probably won't surprise you to learn that my results were consistent with the math, if you make the correct assumptions and set up the problem correctly. After generating one million mothers, then taking the set of mothers who have at least one girl who was born on a Wednesday, I found that 48.18% of them have two girls. If I instead took the set of mothers who have at least one girl with the name Name2 (which is a 1% chance per daughter and equivalent to Julie), then I found that 50.24% of these mothers have two girls. If we just look at the whole set of mothers who have two kids, at least one of which is a girl, then 33.28% of the mothers have two girls. I think your explanation in this video is a lot better than the one in the previous video, and I don't object to this explanation as much as I did to the previous one. I think the nuance of how this information was obtained was noticeably absent from the last video and thats part of the reason that I rejected the conclusion.
Yeah I agree completely and it was definitely some nuanced points the whole time. A few people from the last video made programs and after seeing the problem with the 'paradox' I thought it was interesting to see that even a program won't reflect what is going on if you don't program it EXACTLY how it would reflect reality (in that the programs people made assumed the person TOLD people to step forward who were named Julie or w/e).
Saw the earlier video. Most interesting, but something felt viscerally wrong/incomplete. But this explanation is brilliant and so satisfying. Thank you as I can sleep in peace :)
Awesome video! I literally went to bed rewatching the last video trying to sort out an argument as to why the facts need to be separate. IE My daughter Julie is super smart Also My daughter just had her sweet 16 But I am not sure if both of these facts are about the same girl I just know he could have mentioned his son but didn’t. I subscribed because of these 2 videos you really did a great job with them thanks for the second video it is right on the money as best as I can tell.
You have to be careful with the bet at 9:22. You might walk into a bar full of statisticians (or mathematically educated gamblers), who will only take the bet if their other kid is a girl as well. Or they might be even craftier, and decide that half of those whose other kid is a boy will refuse the bet. Now you are paying 5:4 on a coinflip and you don't even have a way to outsmart them. Worse, it will take you quite some time to figure out what is going on. Getting 15/30, when you expected 20/30 is still within the 95% confidence interval. You might think you are just being unlucky, when you are actually losing 10% of your bet on average.
Now tackle the Monty Hall problem. It is pretty similar. It matters where the information comes from. If the revealed door/goat was randomly chosen, the you get a 50/50. But if Monty selects the goat/door and he knows it is before opening it, then you're odds of winning on a stick is only 1/3. If none of that makes sense, wikipedia can explain the problem in greater detail.
The odds start at 1/3 1/3 1/3, but after you choose one and another door is opened, the other door has a 50% chance of being the goat door, since there was 2 doors from monty to pick from. So if you change, you instead have a 50% chance to win because you are upgrading from your 33% chance door to a 50% chance door. If you stick, you keep the same odds from the start which is 1/3. Or something like that, my diction when it comes to math is horrible but I hope I did a decent job.
@@casualcgrain7835 Actually, switching boosts your odds to 2/3 chance. It is never 50/50 because Monty is aware of what is behind the doors. If his choice was random and some of the time he spoiled the prize, then you're odds would be 50/50. But he knows and he ALWAYS picks a goat. This is why the odds lock in at the start of the game. Imagine I give you a deck of cards face down. You pick one. Now I show you 50 cards that aren't the Ace of spades, should you keep or switch? Obviously you switch, because the chance of you nailing the guess out of the entire deck is really small. So the odds of the ace being the other card is incredibly good. You go from a 1/52 chance to a 51/52 chance. Remember, the entire reason this happens is because the host is aware of what's behind the doors/cards and with 100% accuracy, chooses and removes the misses.
This is how probability paradoxes should be resolved! you can argue the maths all you like, but probability is meant to reflect real life circumstances, so you should conduct experiments (or at least thought experiments like this) to get to the truth of the matter
A problem with your betting idea is that the ones with two girls would be more likely to take the bet since they probably know you're going to guess boy/girl. So you're probably still gonna lose money.
This is what I like about statistics; The fact that it is not just right or wrong if depends on so many other factors. This is the mane reason I NEVER fully believe politicians when you talk about X% is this and Y% is that, I want to know the factors behind the percentages. Kiitos for this follow up video.
At a bar a father (of two kids) approaches you and says: - I have a girl -> P(GG) = 1/2 (Because twice the chance that father with GG says so as compared to a father with a boy and a girl as the latter half a time says I have a boy) - I have a girl named Julie -> P(GG) = 1/2 (Because 1/2 of the times father of two girls talks about the other girl) At a bar you approach a father (of two kids) and ask him: - Do you have a girl? He says yes. -> P(GG) = 1/3 (Because you haven’t messed up with the priors) - Do you have a girl named Julie? If he says yes -> P(GG) = 1/2 (Because twice the chance that fathers with two girls say yes to this question as compared to BG/GB father).
Yep, you got it. I went through the same thing with Monty Hall. I had grasped the reasoning behind the correct answer with that but I then compared it with what at first sight seemed to be an identical scenario -- something to do with three cars, and guessing something about keys -- but which I knew for a fact came out opposite to the doors and goats situation. Then I just sat and pondered what on Earth was making one different from the other. And it quickly became clear. When Monty opens his door, prior to asking you if you want to switch, he never ever opens to the car; it's _always_ a goat (because he knows beforehand). And so the reason the puzzle is puzzling is because we think that all Monty is doing when he opens his door is giving us information that we already had -- i.e. that there is indeed at least one goat behind the unopened doors. -- and it's not clear why that would change the probabilities. But him opening the door and there _always_ being a goat there does a whole lot more than that. It essentially removes 1/3 of the possible situations that could happen if the situation were completely random. That is, it removes all the times when in a truly random situation Monty would open the door and find the _car!_ And so now when I hear someone describe the Monty Hall problem, I listen for the all-important caveat. When they say that Monty opens the door and finds the goat, they must say it so that it's clear it happens like that because Monty _knows_ what is behind each door. Or, to be more precise, it must be clear that when Monty opens his door, there is no chance it will reveal the car. If that caveat is not given -- and it often isn't -- then the probability of the car being behind the other, still-closed door after Monty has opened his door to find a goat is not 2/3 as it is in the usual puzzle scenario. Rather it is only 1/2 and so the answer to the question as to whether you should then switch becomes a resounding "Meh, who cares." And that's what's going on here with your rephrasing. The difference between the original boy/girl scenarios that you describe in the first video, versus the rephrased versions here, is the very same as the difference between a Monty who knows what is behind the doors, versus one who does not.
"and it's not clear why that would change the probabilities" In fact, it _doesn't_ change the probabilities. The probability of the car being behind the contestant's first pick was 1/3 and _remains_ 1/3 after Monty knowingly reveals the goat. If the contestant is not allowed to change his choice, the reveal of the goat doesn't benefit him, nor does it disadvantage him. It's when Monty opens a door without knowing, that the probabilities change: the probability of the car being behnd the contestant's first pick changes from 1/3 to 1/2 if Monty happens to reveal a goat, and from 1/3 to 0 if Monty happens to reveal the car. So if the contestant is not allowed to change his choice, the reveal of the goat benefits hiim.
Yes, OK. I was sooooo annoyed by that prior video because it ignored the self-selected aspect. Now that the selection is external to the person it all lines up just fine.
The given is completely contradictory: It is assumed that a) the name of julie is equally distributed across all girls, and b) that a single family with two children never names both children the same name. Assuming every child has a name, these givens are incompatible. If you drop given a), it is a solvable problem if you build a model around b). To put it clearly: if you have only one girl on the planet named Julie, the chance that Julie has a sister is 50%, and the chance for a family to name a girl julie is (statistically) 0%. If, however, the chance of a family to name a girl julie if they don't already have a julie, is 100%, the chance is 1/3. Every family with a girl will be a family with a julie, and so the the information that a family contains a girl is the same as saying the family contains a julie. All other probabilities for a family to name a girl julie (without having a sister julie) are in between 0% and 100%, and follow a function with values ranging between 50% and 33%. In conclusion, in real life, the paradox will still appy, since the naming probability is always between 0% and 100%. The more unique the name, the closer to 50% the chance of having two girls is, the more common, the closer it is to 33%. This is of course assuming that no families will have the same name, that the name is strictly for girls and that we exclude all families not containing exactly two children.
I think there is still something we are missing! Because if I ask them if they have a daughter the odds are supposed to be 33.3%. If I now guess names until I guess right, the odds are supposed to change to 50%. But as the odds of me getting the name right eventually are 100%, I can also use the formula that leads to 50% before I guessed the name yet and multiply it by 100%, which still equals 50%. But that would mean 33%=50%; which is not true.
The follow up to this video could be about how you were wrong about people taking your bet if it's on their favor, as there's the loss aversion paradox out there about how people would not take bets that were in their favor, not even if the chance is 50% and you'd paid them twice as much as they would lose (that is, you'd fairly be losing money for them, and they could get all your money on the long run, and they wouldn't even accept the first bet), there's a Veritasium bet video about it.
Statisticians should pay atention to Achiles and the turtle. If a "perfect" line of reasoning gives you an obviously wrong answer, plainly it means we made a mistake in our asumptions and/or there's a flaw in what we thought was perfect reasoning.
The main different in asking is, that you ask them about both of their Children. If they just habben to talk about one of their child's which is a girl and you ask them if her name is Julie the Probability does not change, but if you ask if one of their childs is called Julie it's obviously a different question which changes the probability
It is deterministic I was right. It works just like quantum probability that's nuts. The method of observation effects probability at all scales I guess
There's a similar statistical shift when offering the bet as there is when finding out the girls name is Julie. As any father who knows the statistics will know that if you are offering the bet, then you are likely to be guessing he has a son and a daughter. Therefore any (knowledgable) father who accepts the bet will have two daughters, and you will lose your money everytime. --------- Table with stats, neglecting unlikely things. (Sons called Julie, both daughters being Julie, fathers taking bets they believe they would lose, etc.): Event Prob. of 2 G's Has >0 Daughters 1/3 Has >0 D's & 1 is a Julie 1/2 Has >0 D's & will take bet 1/1 Moral of my story is if you go round offering a bet, to guess something the other person already knows. You are very quickly going to lose a lot of money.
Can you make a video on why you're not working in the engineering field.What was your motivation for that. Id like to get an honest and transparent response for this question from a person who knows quite extensively about the field,which in this case is you.Is engineering all that the world portrays it as(like you'll go out there and be working in science mostly,but many just end up with a desk job,not really making a difference to the world but printing profits).Engineering is not what we usually see it as.When you decide to become an engineer its mostly to work in science,make a difference and do something you're passionate about,but end up fulfilling very little of those goals. Would be extremely helpful
Ah yes, probability requires precise wording. Over here in this country, it baffles a lot of students and still separates students who actually do mathematical thinking and students who don't
I just came from the previous video, and I still stand by what I wrote there. In the case of the conversation of two children: "I have two children, one of which is a girl.." . What kind of people will say something like that while having two daughters? I could ask the person when he stopped "Oh so the other is a boy?". According to your statistics half or 33% of the times the answer will be "No the other is a girl". Noone talks like that. "I have two children, one of which is a girl, while the other is a girl. "
I know you're mostly bored with the discussion. But while I definitely agree this isn't a paradox, and this video definitely helped a lot, I do have a clarification that I think is due. It's not really about you asking or them mantioning something by chance. Phrasing that way puts too much emphasis about the power of "asking" and mentioning by "chance" or randomness. You can simplify this and say that in both cases you ask a question: Of all the people in this random sample, who have exactly 2 Children and AT LEAST one daughter, you ask: 1- Do you have a daughter named Julie? (Then you have the chance that out of those who answer "yes", 50% has 2 daughters) Or 2- Tell Me the name of A daughter you have. (Then you have the chance that out of those who nswer "Julie", a 1/3 has 2 daughters) In the second one you the sample you will be making the judgment of probabiity is different. (There are less Julie's and by extention, families in the second one, because ideally you will miss a quarter of all Julies, whose parent's answered with the other daughter's name, since the 500 parents in this example (or 50% of all parents with 2 children, the ones who have a child of each gender) that have a daughter named Julie will answer, while the other parents who have two daughters, who are 25% of the original sample but have 50% of all daughters since they have 2 each, will answer Julie half the time. This actually gives us a different number for any assumed probability for the name Julie). We actualy end up with only 75% the size of our sample in the first case scenario. This idea is better understood in my opinion by a few examples that don't focus on the question or mention, and not even on the yes-or-no or open question. You can have the same effect by asking if the eldest (or youngest) child is called Julie, in which case we end up with 50% the size of the previpus example's first case sample, with 50% chance of them having two daughters. Why is asking any of those questions get us back to the supposedly intuitive idea that the chance for the other child to be a girl is the random chance between a girl and a boy, i.e. 50%? It's actually because the way you are choosing a sample from within a sample is inherently different, and may cancel out the imbalance of the original choice of sample (not that they have 2 children, but that they have at least one daughter) on the ratios involved. Think of it this way: where we get a probability of 50% of having 2 daughters, the first condition of having AT LEAST one daughter, will not affect the ratios in the sample had it not been asked, as parents with two boys NEVER have a Julie as a child in our assumptions. And asking a random group of parents who have 2 children if they have a daughter named Julie, really doesn't tell you anything about the other child. Having AT Least a daughter, does affect the proportions as it includes only 75% of the sample, where 2/3 have ONLY one daughter, and 1/3 has 2 daughters. This is the subtle part of the first episode. Ask that parents with 2 same sex children, or opposite sex children, to step forward and then ask about if they have a child named Julie, will make the probability either a 0% or 100% that the other child of a specific gender. If you ask instead all parents with a child whose name starts with a J, And you don't know anything about the other child if female and male names had the same probability of starting with J. So it's not about any information about the name or knowing something extra, or even asking, it's simply about which group are you sampling by which method: A sample of pairs of binary values; choose a sub-sample by cotaining AT LEAST a specific binary value, you end with a sample are more opposite-value pairs than same-value pairs, double to be precise. Giving you a better chance than 50% to predict the other value. Now take a random sample out of this sub-sample under this one condition: The number of opposite-value pairs is half the the number of same-value pairs. Could you predict the value you don't know with a certainty above 50%? Are there actually more pairs (of those we already know the value of one) of one composition (same-value, opposite-value) than the other? My apologies for the length, but I'm really bad at formalising ideas. On another note, it's fun to think abouot all the different things we have assumed: like having a unisex name, the generalisation to families with any number of children, and how would that make it important to know the distribution of the size of the families in the sample, because of the rule that only one child has the same name in the same family, amking the probability of a name depending on the average family size...etc. I gave up on the generalisations when it entered the half-siblings realm.
I’ll rephrase your question to give a more definite answer. If everyone in the world had their own room with 2 light switches and could pick any configuration of on/off they wanted (let’s assume this is done randomly). Then you went up to one person and asked ‘is at least one of your lights on’ and they say yes. There is then a 1/3 chance that the other light is on.
@@zachstar that's where you're making the mistake. You are guessing the status of the other switch from someone who said at least one is on. You have 50% odds. You are not allowing someone with two off switches to play, and don't appear in the calculations. According to my understanding of the question, the first on switch only indicates whether or not this person can play the game. The second switch indicates the correct answer.
@@zachstar bruh I swear to god this makes no sense and it’s PISSING ME OFF PLEASE HELP ME 😭😭😭😭 if there’s literally TWO light switches... and one of them HAVE TO BE TURNED ON... that means the ONLY other light either has to be ON or OFF.. which is TWO choices... therefore a 50% making it 1/2.... HOW IS IT 2/3 pls HELP ME IM GONNA EXPLODE
Awesome, now I can bet against the guy at the bar that I either asked or he told me that he has 2 children, at least one that is female, and that her name is Julie, assuming that he can't talk about his son if he has one.
Easy way to put many of the comments to rest: flip two coins and mark down if you get double heads, or one heads and one tails. Double tails doesn't count. Repeat enough times that you can satisfyingly get a good percentage of results. Or you could do this and hide the coin tosses, then when at least one coin is heads you have a guesser guess if the other coin is heads or tails. You could even mark each coin as "older" and "younger" or something. I dunno, have some fun with it. The point is that actual experiments are often more convincing then probabilistic theory
… and, how does it work for the Tuesday question, please? I taught that it should not make a difference if you ask or you know. If they say: she was born on a Tuesday, even if they said it randomly, you now know you can exclude the "both of them not born on a Tuesday" case.
Let 1/x= probability that daughter fits criteria given that both daughters can fit said criteria (2x-1)/(4x-1): probability of second child being a girl If you plug in the example of being born on Tuesday: Probability of of the child be born on Tuesday is 1/7, so 1/7=1/x Cross multiply to find x=7, plug x into the equation: (2•7-1)/(4•7-1)= (14-1)(4•7-1)= 13/27 which is approximately 48% As x approaches infinity f(x) approaches .5
Imagine tossing two coins. For each coin there are two outcomes - heads or tails - and the coins are independent. Repeating this, you'll expect to get twice as many mixed pairs (with a head and a tail) as pairs with two heads. Same mechanism as for why in families with two kids you'd expect twice as many with one boy and one girl as families with two girls.
I think a lot of the "controversies" around these probabilities are actually more about semantics (how words are used and what they mean) than they are about mathematics. At least, that's been the case in 100% of the two cases I'm personally aware of. :)
I think it's less about how you get the info, and more about the fact that the follow-up statement is or isn't guaranteed to be talking about the same child: "I have two kids, one of them is a girl, and her name is Julia" => 33.3% "I have two kids, one of them is a girl, and one of them (maybe the same maybe not) is named Julie" => 50%
I am fond of math, but I must say that Random Variables and Statistics is not my favorite. But here is what I think here, probability depends on the information given. But if other information are provided, the probability changes as it makes the "certainty" more apparent. I don't think it depends on whether you asked this information or not, or if it was voluntarily given to you or not. As long as the information is provided, that is what affects the probability.
Alright hopefully this video helps put this all together! As mentioned by wikipedia it was very hard to see why my initial assumptions were wrong. But thank you to everyone in the last video for the discussion about what is really going on within this problem.
Also I don't really talk about the 'born on a Tuesday' part in this video but the same math I talked about would apply (asking would yield a change, whereas someone just mentioning it would keep the probability at 50%).
Nice explanation this time. I seriously thought that your previous vid was april fools joke.
Thanks for making an explanation. I definitely think that the phrasing in the previous video was incorrect, so I'm glad you rephrased it here!
Do more videos like these. They are quite interesting.
now it makes cents do have a girl is so I bet... I don't bet.
The original confusion comes from not stating explicitly the question that brings about the information in each cases. This makes calculating the probability of the given state confusing.
The case where the dad tells us "randomly" about their daughter is the result of asking: "tell me about one of your children". He could have told us about a boy.
The other case is the result of the question: "do you have a daughter?"
He could not have told us about a boy.
Once these starting questions are laid out, the probabilities are easy to calculate.
Same thing with the name:
- "What is the name of your daughter?" He could tell us the name of the other daughter if he has one.
- "Do you have a daughter named Julie?" He could not have told us about the other daughter.
And same thing with the day of birth.
As a side note, DON'T go to bars and start asking guys about their daughters, while putting money down.
There's 99.99 % chance they are gonna kick your ass and 0.01 % chance that they love money more than their daughters. I still don't think 10 dollars is enough.
I should've read the comments before trying it.
And if someone casually mentions they have a daughter, don't just stay there quiet calculating probabilities. It's gonna look weird.
Lol agreed.
@@alansilverio4467 I don't have to because I have seen this video
Just don't go into a bars full of statisticians when you make these bets. Only the ones with two daughters will take you up on the bet and you'll be wondering why you're having such a bad run of luck 😂
Underrated comment
What is the game theory optimal (GTO) behavior of full-knowledge statisticians playing this game -- on both the side of the interrogator/bettor and the bet-acceptors?
But if you know that, you can adjust your strategy to guess they have two daughters! It becomes rock-paper-scissors with extra steps.
@@fdagpigjtrue!
@@weksauceoooh okay so I suppose the only unexploitable strategy for the interrogator would be to randomly guess at 50/50.
The semantics of probability problems has cost me a great deal of letter grades.
The way I made sense of it in my head is this:
When he's talking about his "2 kids, one of whom is a girl", the "one" in question could be one of 2 people if he has two girls. As soon as he says the girl's name, or sees her and says "there she is", he's definitely talking about one specific girl. So the shift comes from a subtle change in meaning of the word "one": It changes from "either of my daughters if I have two daughters, or my only daughter if I have one of each" to "this specific daughter".
To me, this explains the "paradox", and I guess it kind of corresponds nicely to this video's explanation that half the time he'd say "Julie", and half the time he'd mention the "other daughter".
Great video though, it really helped me to start thinking about the paradox in the right way, since I've been aware of the puzzle for a long time and never really felt I'd made proper sense of it.
I came across the prev video today and discussed about the paradox with my friend. We came to a similar conclusion ourselves, but slightly different.
When the name is not told, you are guessing whether the parent has another daughter. But once the parent points/name/specify Julie, and ask whether the other child is a girl, we may think that the question is not changed much but the it is indeed changed.
The question now becomes what is the chance of having another daughter who is not Julie?
@@stonecoldxi9138 If family has 2 children and we see one of those children, a girl, then the chances of other child being girl is 1/3. But if we know some additional data about that girl, for example that it's an older or smarter or better behaved child, then chances are 1/2. Assuming boys and girls are equally smart and well behaved.
But name is not important. If you know girl's name, chances are still 1/3. Unless there is a law that a family of 2 children must name one daughter Julie.
Of the few times I've heard my name in any form of media, this is by far the coolest.
Hehe, nice
Cooler than the song "Oh Julie" by Shakin' Stevens, or the song "Julie July" by Bert Heerink?
Julie… do you have a sister? We need to know.
Oh thank goodness. My visceral anger is gone now. This sits right in my brain.
I still think that there's sth wrong in both vids
@@darkdelphin834 stfu
@@dangerousnigga7023 you could've been raised better
Okay, so if somebody tells you "No, Luke, I am your father", what are the odds that you have a sister?
About the same as Hans Solo having had sex with said sister.
*Never tell me the odds!*
What are the odds that you kissed her?
@@DRAT311 If in Alabama, then 100% chances.
Psywriter
That probability is possible to calculate but a lot of deep research and looking at the census and such to find that probability. Unless we assume there's an equal probability of being born in any state
Well done! You nailed it.
There is so much shallow misinformation about this and related problems. Might be typical for the internet, but strangely enough even some math professors got this one plain wrong. (as was the case for the Monty Hall)
I was going to write you to clarify, but writing is hard and now you already did it in a very nice way.
Note 1: As a physicist, I like to think about it this way:
"asking people to step forward" is a preparation of a "state" or "ensemble" i.e. a set of possibilities
"asking (specific) questions" is a measurement on such a set.
If a problem is given in natural language, we always have to clarify which part is a preparation and which is a measurement!
Note 2: I only noticed one misspeak at 10:12 in this video.
You should have said: "If you --outright-- ask if they have a daughter named X, and you guessed correct, you get 50%"
If you first ask for "having a daughter" and later guess the name, the 33% still stand.
So, again no "jump" in probabilities!
Thank you! Yeah the monty hall problem took me a few hours to fully grasp but this one took me a few days cause the details (in my opinion) were even more subtle. Really interesting problem though and glad I came across it.
@@zachstar Monty Hall is my biggest source of anger. Monty opens a door and shows you a goat. **You then say, "Hey, Monty, do you always open a door that has a goat behind it?" and Monty says "Yes, I like messing with people."** Now you switch. Without knowing that Monty always does that, there isn't enough info.
@@AviDrissman Woah that's actually really interesting. These minor details are enough to change everything about a problem.
@@zachstar Indeed. Every analysis always assumes that because Monty _can_ open a door with a goat, he always does, thus the usual percentage. But the problem as usually stated never says so. Sigh.
Great video, BTW, pointing out the difference. Human language is so... tricky.
@@AviDrissman Why would he open any other door? He won't open the one with the prize, and he won't open yours, there's literally no other door for him to open that isn't a goat.
Outstanding! I knew the comments on your first video would be heated.. But this video explains the subtle details better than any I've seen. Well done!
Actually if the sentence they utter is "I have a daughter," there is roughly a 0% chance they have two daughters, because if they had two daughters they would say "I have two daughters." That's how actual people talk.
@Daniel Copeland You may be the best statistician ever.
Thank you, came to the comments for this. Who the fuck says "I have two kids, one is a daughter." when both are actually daughters.
Daniel Copeland nah, you can say something like “my daughter just graduated high school” which doesn’t rule out you having another daughter and someone can say it
@@tcoren1 "My daughter" and "I have a daughter" are not the same utterance. Only one of them is a complete statement, for one thing.
Yes, the phrase "my daughter" (unlike "I have a daughter") doesn't put an upper constraint on the number of daughters the speaker has. But notice that your sentence does not convey the information that the speaker has two children.
(If they said "I have two kids; my daughter just graduated high school," the natural way to parse that sentence is that the person has exactly one daughter, because otherwise they wouldn't have used the phrase "my daughter" to single out one of the two children. They would have used some other expression that did distinguish the two.)
Face it, this just isn't how people talk. Natural language abhors this particular kind of ambiguity.
Daniel Copeland nobody who has two kids would ever say “I have a daughter” in natural conversation, but they might say “my daughter” it’s true that this doesn’t include the having of two children, that should come from the rest of the conversation
"I have two daughters one of which is a girl whose name is Tuesday."
Probability= 1
Thank you for making these two videos. I saw the first one before my time in the back of an Uber today, and I admit YOUR first part popped into my mind, and I was so happy to see the second part when I got home today. Critical and detailed thinking skills, well, our civilization needs all the practice we can get! And a friendly style with some humor is one of the least threatening ways to go about it. Rock on!!! Oh, and hello from the Bay Area!
Zach looks so happy in this video that he's finally resolved the paradox and the curse has been lifted.
Perfect this makes so much sense now! Thank you for this upload!
The way it dissolves for me is if the person says “one of my children is a girl” vs “my first child is a girl” (or of course, “my second child is a girl” which is the same). The pool of possibilities are GB,BG,GG for the first, and GB,GG for the second. I think all the scenarios are subtle variations on this.
Great video, the way you get the information changes everything.
Random Guy: "Yeah, so my daughter Julie..."
Me: "UGHHH great, thanks a lot! Now I have to go back to a room with 10,000 dads!"
Nonono that'd be fine! You'd have to guess her name for it to not work
Simple foolproof test:
Does the fashion in which you got the information make a difference between boys and girls?
If the info is delivered randomly, no. So it must be 50%.
If you asked specifically for ‘girl’, yes. The prob can be different.
If 'one of which' is a daughter, than there is a 100% chance that the other one is a son, because of otherwise, they would say two of which are daughters
Ofcourse it doesn't come up like this in conversation but someome could mention they have two kids and then later say something like: I saw my daughter today.
In natural language I think I agree with you. I strict mathematical language I'd disagree. It is like the word "or". In natural language, it is one or the other, but not both. In mathematics, it is one, the other, or both.
This is wrong. "One of which is a daughter" can still mean they have two daughters. "Only one of which is a daughter" will yield your 100% result.
Yay, I had a correct explanation for the phrasing error in the previous video!
I think this is weird: I was just explaining this to my better half and she _just. does. not. get. it,_ BUT not for the reasons that you'd think. The usual confusion over this one is the same as for the Month Hall problem: "logically"we conclude the answer to be 50:50 and 33:67 seems counterintuitive - but not my wife...
She starts with: "But you are assuming that boys and girls happen equally often, and that if you already have one girl that the next is equally likely to be a boy or a girl." I explained how spermatogenesis gives almost perfectly equal split, but even if this wasn't the case, for the purposes of the example it isn't important - this is a "perfect" hypothetical - assume 50:50.
Then I talk about the "Julie" issue: "But what if her name is Karen?" I explained that it doesn't matter, the example works regardless. "But what about Esme? That name is very uncommon these days." Still doesn't matter, your just need more families to capture enough 'Esmes'. "Oooh, but that is different - you didn't say anything about getting more families."
So I explain again (and again and again) about this being an idealized scenario. As such, where it is not stated, the assumption is that any confounding variables are perfectly average. i.e. the two groups of families (BG and GG) are indistinguishable, that having one girl doesn't change the way the parents would choose a new daughter's name - independent. "But you didn't say that!"
I have degrees in physics and engineering. My wife is a PhD academic who uses *_ QUALITATIVE_* methods in her research. Apparently basic "quant" doesn't compute to a "qual" researcher. Exactly ZERO of the normal assumptions I make (and I assume most others here too) are assumptions that she makes. Bizarre!!
funny stuff. but yeah, I agree agree that this girl girl problem has the same common sense paradox as the Monty Hall problem. If a drunk runs on stage and opens a "random" door and Monty announces "wild stuff! I was about to open a door to help you but now I'll skip that!" you just lost your 2 to 1 advantage in switching doors, you get 50/50 either way.
oops, should have added: because Monty always reveals a goat, the drunk could reveal goat or car.
lol she's really getting hung up on the name. The name is not actually what matters. It's the act of specifying a girl at all at the beginning that changes the probability.
You could say, "I have two daughter, one of which is...standing right there." This changes the probability in the same way because you are specifying a daughter that is 100% exists
The answer is, again, as I believe, not 33.3% because we know that all of humanity is unique and cannot selectively pick and choose. Girl A and Girl B can be interchanged to find a 50% probability in your original problem, and labeling them is necessary because they are unique and it correctly identifies the probable situation.
By nature of language and implying context, "I have two kids, one is a girl" means the other is a boy 99% of the time :D
yes, but "one is a girl" and "at least one of which is a girl" are different, though one could argue that "one is X" is ambiguous and could mean both "one (and exactly one) is X" and "one is X (but I leave the other one totally open)"
Essentially for those who didn't get it, the dads who have 2 daughters might not actually tell you about Julie unless you specifically ask.
I like this follow up. The way I like to think about it is whether you’ve received new information. Asking gets you new information, whereas when you just let people talk, they’ll tell you things randomly in a way that reflects underlying probabilities and you won’t get new information
What do you mean, "you won't get new information"? You do get new information, just not information that may be useful in Zach's particular trick of scamming people out of money.
@@yurenchu i think he means to search for the info yourself, or to wait for the chance of them to speak about the info
@@limonick Nah, I think his understanding (as is the case with most people's understanding) of "new information" is just faulty. When you approach a guy at the bar whom you know has two kids, and the guy hasn't spoken yet, then the odds of him having two daughters versus him having only one daughter are 1:2 ; he's twice as likely to have a daughter and a son, compared to having two daughters. When you ask "Do you have at least one daughter?" and he answers "Yes", then you do receive a bit of new information because now you know for certain that he doesn't have two sons, but the odds between "two daughters" and "one daughter and one son" haven't changed; they remain 1:2 .So in that sense, you actually _didn't_ receive "new information" regarding the odds between "two daughters" and "one daughter and one son".
Whereas if the guy spontaneously says "I have at least one daughter" (or "I have a daughter who likes [insert some hobby here]", or something alike), then the odds do change, from 1:2 to 1:1 (or 50:50); so then you did receive "new information" that changed the odds. Only this type of information is less suited for the particular exploitation scam (money bet) that this video creator was thinking of.
It's similar to the case of the Monty Hall Problem. In that problem, it makes a difference if Marty (the game show host) knows what's behind the doors and always reveals a goat, or if he doesn't know himself what's behind the doors and the fact that he revealed the goat was just coincidence (because there was also the possibility that he would have revealed the car). In the first case, it's attractive for the contestant to switch to the other remaining door because that door is twice as likely to be the door with the car, as the door of the contestant's first pick. In the second case, it doesn't matter if the contestant switches or stays, because both the other door and the door of the contestant's first pick have a 50% likelihood to be the door with the car.
Some people then word this concept as "If the host doesn't know what's behind the doors and he happens to reveal the goat by mere chance, then you didn't receive new information. It's only when the host does know what's behind the doors and always opens a door with a goat that you receive new information, which changes the odds." Well, no. In fact, it's the exact opposite. If a knowing host revealed the goat by design, then the probability that the door of the contestant's first pick is the door with the car hasn't changed: it was 1/3 (or 33.3333...%), and it remained 1/3 . Whereas if an unknowing host revealed the goat by mere coincidence, then the probability does change: from 1/3 to 1/2 (= 50%); in other words, you/the contestant did receive "new information" that improved the chance of the door of the contestant's first pick significantly.
The key thing is: "new information" doesn't always mean "gaining an advantage". New information may actually increase _disadvantage_ . (For example, when Marty knowingly opens the door with the car and says "You lose", then you are receiving new information that changes the probablity of your first picked door from 1/3 to 0 .) However, many people, including (in my view) the commenter above, have trouble making that distinction.
You nailed it! Excellent follow-up video.
'Joke's on you! I actually have three children!'
Another example of this comes from the card game bridge, where you are dealt a hand of 13 cards. If at least one of your cards is an ace, the probability you have at least two aces is slightly less than 1/2. If at least one of your cards is the ace of clubs, the probability you have at least two aces is slightly greater than 1/2.
7:08 Dad with 2 daughters: "I have 2 kids, one of which is a girl"
Thanks for the conclusion! Very helpful
This probability stuff is really hard to get ones head around, so it's nice to see a thorough explanation.
I think some of the Markov chain examples might benefit from an additional representation as a probability tree to add another perspective, if you decide to make more videos on such subjects.
One note about that betting game:
Who's more likely to accept your bet?
- The strangers who are not sure why you're asking them about their daughters and making bets about them?
- Or the stranger who thinks they already know what your angle is?
Thank you. After being presented with the paradox of how adding days to the the mix effects the probability I spent a lot of time thinking about it and thought that probability must be broken. This video made it quite clear to me.
You've got it! Very good. Nice wrap up.
It's actually easy! Think of it as flipping two coins instead . If I say that at least one of my coins came up heads there are three possible states: heads/heads, heads/tails, tails/heads. BUT if I also say that only a single coin which came heads has an X marking it, then there are FOUR possible states: xheads/heads, heads/xheads, xheads/tails, tails/xheads. By telling you that ONLY one of the coins that came up heads has an identifying attribute, we've made it possible to differentiate between both head/head combinations, which changes the number of identifiable states, and thus the probability.
The reason this seems so counterintuitive with the human example is that we "assume" that we can tell two people apart, but the statistics don't make that assumption which is why the probability changes once one of the children becomes identifiable and we move from three possible states to four: Julie/girl, girl/Julie, Julie/boy, boy/Julie.
oh my god thank you
What if the coin lands on edge? Lol
What is your exact model for the experiment? Someone is throwing two coins at the same time where one is marked with an X on the heads side?
@@TheTim466 I'm just using the pair of coins to demonstrate each valid state that they might be in, given specific information that we have. We can assume that the chance of any single coin alone being heads is 50/50. If all we know is that at least one coin is heads then there are three possible states and only one involves the second coin being heads so the chances are 1/3. If we additionally know that the coin which is heads has an X on it (the coin has been "named" just like our Julie example) then there are four possible states for the two coin pair to be in, and two of them involve both coins being heads so the chances of coin two being heads is 50/50
This is a very powerful and helpful response. Thank you.
If a proof is a reasoned argument that helps people to accept something, then this seems to be a good example of how human psychology and wording can make a difference to the quality of the proof.
Really clever how you managed to correct the other video without drawing too much attention to what you are doing: correcting it ;-)
This was a great video. Thanks for making this. I feel like I learned a lot.
it's the difference between saying
"I have 2 daughters and I'm randomly thinking of one of them"
vs
"I have 2 daughters but I'm being directed to think about a specific one"
And similarly
"I have a boy and a girl and am randomly thinking about one of them"
vs
"I have a boy and a girl, and am being directed to think about the daughter"
This was a good video. Thank you.
This is amazing!
I guess the best explanation is to look at who comes up with the data: If you pick the gender, you're performing a postselection, which can introduce correlations to the dataset (you're excluding 50% of cases with matching genders, so the remaining cases with matching genders are fewer in total!). On the other hand, learning data from all samples (gender-of-random) cannot change independent data in the same samples (gender-of-other).
Now, the Julia case is interesting in that if you're postselecting on a property that has zero chance of occurring (each of infinitely many names is equally likely), you end up with no samples and you can't do probabilites over that, so you have to take the limiting case (each of finitely many names is equally likely, but what if we keep adding names). That you can't introduce correlations if you remove almost no samples is straightforward. That you can't introduce correlations in what's left if you remove almost all samples in this specific way is somewhat interesting, but I do suspect that once you define "this specific way" rigorously enough, the non-correlation falls out naturally (as it does in the next paragraph).
But the cleanest way to retrieve our conditional probabilities is to use geometry: If we sort the names by gender first and then by alphabet, we can place all samples in a square. Then {first-is-girl} is a rectangle with two equal parts, {GB} and {GG}. {any-is-girl} is a triomino composed of {GB}, {BG} and {GG}, so {GG | any-G} is 33%. {first-is-julia} now becomes a line in that square, and {any-is julia} becomes a union of two lines. And each of these lines is split evenly by the boy/girl line.
Thank you for taking my comment seriously in the previous video, thus making this one.
I feel like the simplest explanation of this is that the verbal informing of the name isn't what changes the probability. It's the fact that you're restricting the set of available parents you _could_ be talking to to ones that have daughters named Julie, and _that_ is what makes the probability 50%. In the general population, the odds are 2:1 (for the reasons early in the first video... G:G, G:B, B:G). In the subset of people with a daughter named Julie (or any other same name across the set being considered), the odds are 1:1 (again, for the reasons explained in your first video.) The explanations for each probability are spot on. It's the attribution of the change to the "telling" that causes the apparent 'paradox'.
It's a super interesting problem to think about from the perspective of... well... perspective. But from a probabilistic standpoint, it's pretty easily explained in the first few minutes of the original video if you take the focus off of the "telling" and put it on the selection process in the first place.
(For example, you can write a pretty simple program that demonstrates the general 2:1 ratio with randomized data. When you add an additional conditional that checks if they have a daughter named Julie before even prompting a guess, the results will become a roughly even split of 1:1.)
Also I'm definitely not going out to a bar and doing this.
Here's one assumption to make, which I've seen in some versions of the problem and wasn't discussed in either MajorPrep video: What if you assume the father says, "I have two children, at least one of whom is a (boy or girl)", regardless of what type of kids he has? Obviously, if he has two boys, he'll say "boy", and if he has two girls, he'll say "girl". Further assume that, if he has one of each, he'll flip a coin to decide if he'll say "boy" or "girl" (50-50 chance). In this case, if the father says "...at least one of whom is a girl", the chance that both children are girls will be 50%.
So basically testimonials are too random for our brains to settle with but we're totally okay with surveyed responses
This surprisingly made perfect sense to me.
Thank God you made this follow up, I was fuming after watching the original
An easy explanation to this is if information is presented about one child, it depends on how it was presented and by who and with what knowledge and intention. If you hear the name of one of the children who happens to be a girl, the odds of two daughters is now 1/2. However, if someone intentionally gave you the name of a daughter to keep you guessing the number of daughters, it’s still 1/3. It’s the same information but presented in different ways and therefore has a different effect on the odds. It’s like the Monty Hall problem had the one door revealed by the host who has knowledge of the correct door and a purpose of concealing it. Hence the chances of getting the right door by switching was 2/3. However, if that same door were revealed by a random audience member with no knowledge of what’s behind each door, the chances of getting the right door by switching would be 1/2. Again, same information, but different circumstances. Hence, different odds
We should notice that between the two cases of "let US filter for properties we are interested in" and "someone has come to us declaring a property we are NOW interested in" there is already a pre-selection going on, and if that person has to fit multiple properties, the more they fit them, the larger the chance they come to us with that declaration in the first place. In the latter case, the other side decided the metrics we are interested in, which they by definition fit.
If a WW1+WW2 veteran who is 120 years old now, was an astronaut, knows 20 instruments and 40 languages comes to me with all that information, the chance of all that occurring is 100% AFTER THE FACT, but if we are interested from the getgo whether someone like that even exists, well, good luck with that. That's the level of distortion that can be caused in the probabilities depending on what the initial group we look at is.
Ok. I think I've got it in an intuitive way: it's really the difference between the odds of a pair and the odds of a particular child. Siblings genders are (we'll presume) independent, which is how we get the 50/50. But when you're talking about a specific sibling, that swaps the perspective and it really does become like the Monty Hall problem.
Ok, this video makes so much more sense
If I ask a guy in the bar if he has a daughter named Julie (and he does), he's not going to say yes. He's going to start a fight with me, or walk away, or get really scared.
He's not going to go "I actually do. You want to guess the gender of my other child?"
Going back to the situation in the first video, where the guy says "I have a daughter called Julie."
You have 10,000 families and the chance of a girl being called Julie is 1 in 100. Of the 5000 BG/GB families, 50 will have a girl called Julie. Of the 2500 families with two girls, 25 will have a first-born called Julie. These parents will not call their second daughter Julie. So there remain 2475 families who may call their second girl Julie, so there will be 24.75 second daughters called Julie.
To get even numbers, start with 1,000,000 families, with 500,000 BG/GB families and 250,000 GG families. From the BG/GB families we have 5,000 Julies. From the first-born of the GG families we have 2,500 Julies. But the families who have already named their first daughter Julie will not so name their second. That leaves 247,500 families, of which 2,475 name their daughter Julie. So the chance that a family with two children, one of whom is named Julie, will have 2 daughters is 2475/2500+2475, or .4987 (if my arithmetic is correct.)
You are exactly right. This was my main issue from the original video - the lack of a perfect probability. I had to assume that Zach was being lazy and just rounded to %50.
Excellent job sir!
That means if I do the extra work and guess the name of the Girl, my chances of winning would go down? Thats not nice...
Here ,take an apple🍎
🍏This one is from shinigami realm.. don't eat it
Thanks for an interesting follow up. Following our discussion on the previous video, I was motivated to write a Python script where I simulated some scenarios. I created a large number of "Mother" objects, each of which has a random number of children (I weighted each of these according to actual statistics on how many children American women tend to have each, though of course there was some rounding off for simplification) and each child is assigned a random day of the week as their birth day. I then used 200 different names ("Name1" through "Name200", where odd names are assigned to males and even names are assigned to females).
It probably won't surprise you to learn that my results were consistent with the math, if you make the correct assumptions and set up the problem correctly. After generating one million mothers, then taking the set of mothers who have at least one girl who was born on a Wednesday, I found that 48.18% of them have two girls. If I instead took the set of mothers who have at least one girl with the name Name2 (which is a 1% chance per daughter and equivalent to Julie), then I found that 50.24% of these mothers have two girls. If we just look at the whole set of mothers who have two kids, at least one of which is a girl, then 33.28% of the mothers have two girls.
I think your explanation in this video is a lot better than the one in the previous video, and I don't object to this explanation as much as I did to the previous one. I think the nuance of how this information was obtained was noticeably absent from the last video and thats part of the reason that I rejected the conclusion.
Yeah I agree completely and it was definitely some nuanced points the whole time. A few people from the last video made programs and after seeing the problem with the 'paradox' I thought it was interesting to see that even a program won't reflect what is going on if you don't program it EXACTLY how it would reflect reality (in that the programs people made assumed the person TOLD people to step forward who were named Julie or w/e).
For me this is the most disconnected video in this lovely channel
Saw the earlier video. Most interesting, but something felt viscerally wrong/incomplete. But this explanation is brilliant and so satisfying. Thank you as I can sleep in peace :)
Awesome video! I literally went to bed rewatching the last video trying to sort out an argument as to why the facts need to be separate. IE
My daughter Julie is super smart
Also
My daughter just had her sweet 16
But I am not sure if both of these facts are about the same girl I just know he could have mentioned his son but didn’t.
I subscribed because of these 2 videos you really did a great job with them thanks for the second video it is right on the money as best as I can tell.
I somehow feel like this bears on collapse of a wave function, asking where a particle is..
You have to be careful with the bet at 9:22. You might walk into a bar full of statisticians (or mathematically educated gamblers), who will only take the bet if their other kid is a girl as well. Or they might be even craftier, and decide that half of those whose other kid is a boy will refuse the bet. Now you are paying 5:4 on a coinflip and you don't even have a way to outsmart them.
Worse, it will take you quite some time to figure out what is going on. Getting 15/30, when you expected 20/30 is still within the 95% confidence interval. You might think you are just being unlucky, when you are actually losing 10% of your bet on average.
Now tackle the Monty Hall problem.
It is pretty similar. It matters where the information comes from. If the revealed door/goat was randomly chosen, the you get a 50/50. But if Monty selects the goat/door and he knows it is before opening it, then you're odds of winning on a stick is only 1/3.
If none of that makes sense, wikipedia can explain the problem in greater detail.
The odds start at 1/3 1/3 1/3, but after you choose one and another door is opened, the other door has a 50% chance of being the goat door, since there was 2 doors from monty to pick from. So if you change, you instead have a 50% chance to win because you are upgrading from your 33% chance door to a 50% chance door. If you stick, you keep the same odds from the start which is 1/3. Or something like that, my diction when it comes to math is horrible but I hope I did a decent job.
@@casualcgrain7835 Actually, switching boosts your odds to 2/3 chance.
It is never 50/50 because Monty is aware of what is behind the doors. If his choice was random and some of the time he spoiled the prize, then you're odds would be 50/50. But he knows and he ALWAYS picks a goat. This is why the odds lock in at the start of the game.
Imagine I give you a deck of cards face down. You pick one. Now I show you 50 cards that aren't the Ace of spades, should you keep or switch? Obviously you switch, because the chance of you nailing the guess out of the entire deck is really small. So the odds of the ace being the other card is incredibly good. You go from a 1/52 chance to a 51/52 chance.
Remember, the entire reason this happens is because the host is aware of what's behind the doors/cards and with 100% accuracy, chooses and removes the misses.
Guess my memory of it has eroded, thanks for the refresher. I understood the basic principle just forgot the numbers.
This is how probability paradoxes should be resolved! you can argue the maths all you like, but probability is meant to reflect real life circumstances, so you should conduct experiments (or at least thought experiments like this) to get to the truth of the matter
A problem with your betting idea is that the ones with two girls would be more likely to take the bet since they probably know you're going to guess boy/girl. So you're probably still gonna lose money.
This is what I like about statistics; The fact that it is not just right or wrong if depends on so many other factors. This is the mane reason I NEVER fully believe politicians when you talk about X% is this and Y% is that, I want to know the factors behind the percentages.
Kiitos for this follow up video.
VeryEvilPettingZoo OK I agree with you
I'm just old and a little cynic so I think "they might make mistakes' 😃😃😃😃😃
"Lies, damned lies and statistics" -- Benjamin Disraeli
What are kittos?? You mean kudos?
@@vidicate3963 I think he also meant "main", not "mane", but I could be "lion" about that. :-)
Key point: If a parent has two daughters, one of which is named Julie, and tells you about one daughter at random, half the time it WON'T be Julie.
At a bar a father (of two kids) approaches you and says:
- I have a girl -> P(GG) = 1/2
(Because twice the chance that father with GG says so as compared to a father with a boy and a girl as the latter half a time says I have a boy)
- I have a girl named Julie -> P(GG) = 1/2
(Because 1/2 of the times father of two girls talks about the other girl)
At a bar you approach a father (of two kids) and ask him:
- Do you have a girl? He says yes. -> P(GG) = 1/3
(Because you haven’t messed up with the priors)
- Do you have a girl named Julie? If he says yes -> P(GG) = 1/2
(Because twice the chance that fathers with two girls say yes to this question as compared to BG/GB father).
Yep, you got it. I went through the same thing with Monty Hall. I had grasped the reasoning behind the correct answer with that but I then compared it with what at first sight seemed to be an identical scenario -- something to do with three cars, and guessing something about keys -- but which I knew for a fact came out opposite to the doors and goats situation. Then I just sat and pondered what on Earth was making one different from the other. And it quickly became clear. When Monty opens his door, prior to asking you if you want to switch, he never ever opens to the car; it's _always_ a goat (because he knows beforehand).
And so the reason the puzzle is puzzling is because we think that all Monty is doing when he opens his door is giving us information that we already had -- i.e. that there is indeed at least one goat behind the unopened doors. -- and it's not clear why that would change the probabilities. But him opening the door and there _always_ being a goat there does a whole lot more than that. It essentially removes 1/3 of the possible situations that could happen if the situation were completely random. That is, it removes all the times when in a truly random situation Monty would open the door and find the _car!_
And so now when I hear someone describe the Monty Hall problem, I listen for the all-important caveat. When they say that Monty opens the door and finds the goat, they must say it so that it's clear it happens like that because Monty _knows_ what is behind each door. Or, to be more precise, it must be clear that when Monty opens his door, there is no chance it will reveal the car. If that caveat is not given -- and it often isn't -- then the probability of the car being behind the other, still-closed door after Monty has opened his door to find a goat is not 2/3 as it is in the usual puzzle scenario. Rather it is only 1/2 and so the answer to the question as to whether you should then switch becomes a resounding "Meh, who cares."
And that's what's going on here with your rephrasing. The difference between the original boy/girl scenarios that you describe in the first video, versus the rephrased versions here, is the very same as the difference between a Monty who knows what is behind the doors, versus one who does not.
"and it's not clear why that would change the probabilities"
In fact, it _doesn't_ change the probabilities. The probability of the car being behind the contestant's first pick was 1/3 and _remains_ 1/3 after Monty knowingly reveals the goat. If the contestant is not allowed to change his choice, the reveal of the goat doesn't benefit him, nor does it disadvantage him.
It's when Monty opens a door without knowing, that the probabilities change: the probability of the car being behnd the contestant's first pick changes from 1/3 to 1/2 if Monty happens to reveal a goat, and from 1/3 to 0 if Monty happens to reveal the car. So if the contestant is not allowed to change his choice, the reveal of the goat benefits hiim.
There is a notation for this.
P(A/B) which is the probability of the event A knowing that the event B happened.
P(A/B)=P(A and B)/P(B)
Yes, OK. I was sooooo annoyed by that prior video because it ignored the self-selected aspect. Now that the selection is external to the person it all lines up just fine.
A room with 10.000 dads is not a room, it's a stadium!
You've solved the observer paradox in quantum mechanics.
How
Ight I’m ruining 1000 dads nights to conduct this random experiment
The given is completely contradictory: It is assumed that a) the name of julie is equally distributed across all girls, and b) that a single family with two children never names both children the same name.
Assuming every child has a name, these givens are incompatible.
If you drop given a), it is a solvable problem if you build a model around b).
To put it clearly: if you have only one girl on the planet named Julie, the chance that Julie has a sister is 50%, and the chance for a family to name a girl julie is (statistically) 0%.
If, however, the chance of a family to name a girl julie if they don't already have a julie, is 100%, the chance is 1/3. Every family with a girl will be a family with a julie, and so the the information that a family contains a girl is the same as saying the family contains a julie.
All other probabilities for a family to name a girl julie (without having a sister julie) are in between 0% and 100%, and follow a function with values ranging between 50% and 33%.
In conclusion, in real life, the paradox will still appy, since the naming probability is always between 0% and 100%. The more unique the name, the closer to 50% the chance of having two girls is, the more common, the closer it is to 33%.
This is of course assuming that no families will have the same name, that the name is strictly for girls and that we exclude all families not containing exactly two children.
"I have two kids, one is a girl."
"So the other is a boy."
"Eh, 50-50."
I think there is still something we are missing! Because if I ask them if they have a daughter the odds are supposed to be 33.3%. If I now guess names until I guess right, the odds are supposed to change to 50%. But as the odds of me getting the name right eventually are 100%, I can also use the formula that leads to 50% before I guessed the name yet and multiply it by 100%, which still equals 50%. But that would mean 33%=50%; which is not true.
Sounds like a carnival game trick
The follow up to this video could be about how you were wrong about people taking your bet if it's on their favor, as there's the loss aversion paradox out there about how people would not take bets that were in their favor, not even if the chance is 50% and you'd paid them twice as much as they would lose (that is, you'd fairly be losing money for them, and they could get all your money on the long run, and they wouldn't even accept the first bet), there's a Veritasium bet video about it.
Statisticians should pay atention to Achiles and the turtle. If a "perfect" line of reasoning gives you an obviously wrong answer, plainly it means we made a mistake in our asumptions and/or there's a flaw in what we thought was perfect reasoning.
The main different in asking is, that you ask them about both of their Children. If they just habben to talk about one of their child's which is a girl and you ask them if her name is Julie the Probability does not change, but if you ask if one of their childs is called Julie it's obviously a different question which changes the probability
It is deterministic I was right. It works just like quantum probability that's nuts. The method of observation effects probability at all scales I guess
There's a similar statistical shift when offering the bet as there is when finding out the girls name is Julie.
As any father who knows the statistics will know that if you are offering the bet, then you are likely to be guessing he has a son and a daughter.
Therefore any (knowledgable) father who accepts the bet will have two daughters, and you will lose your money everytime.
---------
Table with stats, neglecting unlikely things. (Sons called Julie, both daughters being Julie, fathers taking bets they believe they would lose, etc.):
Event Prob. of 2 G's
Has >0 Daughters 1/3
Has >0 D's & 1 is a Julie 1/2
Has >0 D's & will take bet 1/1
Moral of my story is if you go round offering a bet, to guess something the other person already knows. You are very quickly going to lose a lot of money.
Hence you should offer the bet to statistically qualified fathers, and guess girl girl
Can you make a video on why you're not working in the engineering field.What was your motivation for that.
Id like to get an honest and transparent response for this question from a person who knows quite extensively about the field,which in this case is you.Is engineering all that the world portrays it as(like you'll go out there and be working in science mostly,but many just end up with a desk job,not really making a difference to the world but printing profits).Engineering is not what we usually see it as.When you decide to become an engineer its mostly to work in science,make a difference and do something you're passionate about,but end up fulfilling very little of those goals.
Would be extremely helpful
Ah yes, probability requires precise wording. Over here in this country, it baffles a lot of students and still separates students who actually do mathematical thinking and students who don't
I just came from the previous video, and I still stand by what I wrote there. In the case of the conversation of two children: "I have two children, one of which is a girl.." . What kind of people will say something like that while having two daughters? I could ask the person when he stopped "Oh so the other is a boy?". According to your statistics half or 33% of the times the answer will be "No the other is a girl". Noone talks like that. "I have two children, one of which is a girl, while the other is a girl. "
I know you're mostly bored with the discussion.
But while I definitely agree this isn't a paradox, and this video definitely helped a lot, I do have a clarification that I think is due.
It's not really about you asking or them mantioning something by chance. Phrasing that way puts too much emphasis about the power of "asking" and mentioning by "chance" or randomness.
You can simplify this and say that in both cases you ask a question:
Of all the people in this random sample, who have exactly 2 Children and AT LEAST one daughter, you ask:
1- Do you have a daughter named Julie? (Then you have the chance that out of those who answer "yes", 50% has 2 daughters)
Or
2- Tell Me the name of A daughter you have. (Then you have the chance that out of those who nswer "Julie", a 1/3 has 2 daughters)
In the second one you the sample you will be making the judgment of probabiity is different. (There are less Julie's and by extention, families in the second one, because ideally you will miss a quarter of all Julies, whose parent's answered with the other daughter's name, since the 500 parents in this example (or 50% of all parents with 2 children, the ones who have a child of each gender) that have a daughter named Julie will answer, while the other parents who have two daughters, who are 25% of the original sample but have 50% of all daughters since they have 2 each, will answer Julie half the time. This actually gives us a different number for any assumed probability for the name Julie). We actualy end up with only 75% the size of our sample in the first case scenario.
This idea is better understood in my opinion by a few examples that don't focus on the question or mention, and not even on the yes-or-no or open question.
You can have the same effect by asking if the eldest (or youngest) child is called Julie, in which case we end up with 50% the size of the previpus example's first case sample, with 50% chance of them having two daughters.
Why is asking any of those questions get us back to the supposedly intuitive idea that the chance for the other child to be a girl is the random chance between a girl and a boy, i.e. 50%?
It's actually because the way you are choosing a sample from within a sample is inherently different, and may cancel out the imbalance of the original choice of sample (not that they have 2 children, but that they have at least one daughter) on the ratios involved.
Think of it this way: where we get a probability of 50% of having 2 daughters, the first condition of having AT LEAST one daughter, will not affect the ratios in the sample had it not been asked, as parents with two boys NEVER have a Julie as a child in our assumptions. And asking a random group of parents who have 2 children if they have a daughter named Julie, really doesn't tell you anything about the other child. Having AT Least a daughter, does affect the proportions as it includes only 75% of the sample, where 2/3 have ONLY one daughter, and 1/3 has 2 daughters. This is the subtle part of the first episode.
Ask that parents with 2 same sex children, or opposite sex children, to step forward and then ask about if they have a child named Julie, will make the probability either a 0% or 100% that the other child of a specific gender. If you ask instead all parents with a child whose name starts with a J, And you don't know anything about the other child if female and male names had the same probability of starting with J.
So it's not about any information about the name or knowing something extra, or even asking, it's simply about which group are you sampling by which method:
A sample of pairs of binary values; choose a sub-sample by cotaining AT LEAST a specific binary value, you end with a sample are more opposite-value pairs than same-value pairs, double to be precise. Giving you a better chance than 50% to predict the other value.
Now take a random sample out of this sub-sample under this one condition:
The number of opposite-value pairs is half the the number of same-value pairs.
Could you predict the value you don't know with a certainty above 50%?
Are there actually more pairs (of those we already know the value of one) of one composition (same-value, opposite-value) than the other?
My apologies for the length, but I'm really bad at formalising ideas.
On another note, it's fun to think abouot all the different things we have assumed: like having a unisex name, the generalisation to families with any number of children, and how would that make it important to know the distribution of the size of the families in the sample, because of the rule that only one child has the same name in the same family, amking the probability of a name depending on the average family size...etc.
I gave up on the generalisations when it entered the half-siblings realm.
Ah the good old ambiguous question. Have you checked out the cords through a circle paradox (I think it's numberphile) that's a head scratcher.
What if you had two light switches, and are told that at least one is turned on. What is the chance that the other light switch is turned on also?
I’ll rephrase your question to give a more definite answer. If everyone in the world had their own room with 2 light switches and could pick any configuration of on/off they wanted (let’s assume this is done randomly). Then you went up to one person and asked ‘is at least one of your lights on’ and they say yes. There is then a 1/3 chance that the other light is on.
@@zachstar that's where you're making the mistake. You are guessing the status of the other switch from someone who said at least one is on. You have 50% odds. You are not allowing someone with two off switches to play, and don't appear in the calculations.
According to my understanding of the question, the first on switch only indicates whether or not this person can play the game. The second switch indicates the correct answer.
@@zachstar bruh I swear to god this makes no sense and it’s PISSING ME OFF PLEASE HELP ME 😭😭😭😭 if there’s literally TWO light switches... and one of them HAVE TO BE TURNED ON... that means the ONLY other light either has to be ON or OFF.. which is TWO choices... therefore a 50% making it 1/2....
HOW IS IT 2/3 pls HELP ME IM GONNA EXPLODE
@@zachstar I meant 1/3 ** HOWW
Awesome, now I can bet against the guy at the bar that I either asked or he told me that he has 2 children, at least one that is female, and that her name is Julie, assuming that he can't talk about his son if he has one.
Easy way to put many of the comments to rest: flip two coins and mark down if you get double heads, or one heads and one tails. Double tails doesn't count. Repeat enough times that you can satisfyingly get a good percentage of results.
Or you could do this and hide the coin tosses, then when at least one coin is heads you have a guesser guess if the other coin is heads or tails.
You could even mark each coin as "older" and "younger" or something.
I dunno, have some fun with it. The point is that actual experiments are often more convincing then probabilistic theory
… and, how does it work for the Tuesday question, please?
I taught that it should not make a difference if you ask or you know.
If they say: she was born on a Tuesday, even if they said it randomly, you now know you can exclude the "both of them not born on a Tuesday" case.
This is soooo much clearer than the previous video. WEll done because the previous one destroyed my brain :-D
But your brain is now comfortably numb that is why you feel this explanation is better.
Let 1/x= probability that daughter fits criteria given that both daughters can fit said criteria
(2x-1)/(4x-1): probability of second child being a girl
If you plug in the example of being born on Tuesday:
Probability of of the child be born on Tuesday is 1/7, so 1/7=1/x
Cross multiply to find x=7, plug x into the equation:
(2•7-1)/(4•7-1)=
(14-1)(4•7-1)=
13/27 which is approximately 48%
As x approaches infinity f(x) approaches .5
I still don’t see why it matters to probability if it’s Girl Boy or Boy Girl?? To get 33.3% rather than just 50%
Imagine tossing two coins. For each coin there are two outcomes - heads or tails - and the coins are independent. Repeating this, you'll expect to get twice as many mixed pairs (with a head and a tail) as pairs with two heads. Same mechanism as for why in families with two kids you'd expect twice as many with one boy and one girl as families with two girls.
im still so confused.
I think a lot of the "controversies" around these probabilities are actually more about semantics (how words are used and what they mean) than they are about mathematics. At least, that's been the case in 100% of the two cases I'm personally aware of. :)
I think it's less about how you get the info, and more about the fact that the follow-up statement is or isn't guaranteed to be talking about the same child:
"I have two kids, one of them is a girl, and her name is Julia" => 33.3%
"I have two kids, one of them is a girl, and one of them (maybe the same maybe not) is named Julie" => 50%
Conditional probability is a cruel mistress. It'd have been easier if you explained it with Bayes' law.
I am fond of math, but I must say that Random Variables and Statistics is not my favorite.
But here is what I think here, probability depends on the information given. But if other information are provided, the probability changes as it makes the "certainty" more apparent. I don't think it depends on whether you asked this information or not, or if it was voluntarily given to you or not. As long as the information is provided, that is what affects the probability.