THANK YOU for discussing probability WITHOUT mentioning statistics! Probability and statistics are often lumped together as if they formed one big word. Addressing each separately is necessary before we can realize the immense power inherent in using them together.
They are very closely related though. I mean statistics is basically applied probability. You use the tools developed in probability theory almost exclusively. And probability theory without statistics would be half as interesting, as you wouldn't have an application of it. But I guess mathematics is also closely related to physics and IT in that sense, so I see where you are coming from. I wonder though if that episode was understandable without already knowing about probability theory. The explanation of random variables for example was extremely short. I doubt anyone could understand random variables from that short comment.
In what sense is statistic the inverse of probability? That would imply that they cancel out in some sense. Which doesn't make any sense so it seems like you just threw words around. While statistics is heavily dependent on probability theory. You use the models and axioms of probability theory like Random variables, stochastic processes, etc. And you use the theorems of probability theory like the different central limit theorems, law of large numbers, Bayes, measurable mappings of independent random variables are still independent, etc. to base all your theorems of statistic on. Or you can justify using the central limit theorem that you assume something is normal distributed. Everything in statistics is based on probability theory. You could probably say statistics is to probability theory, what probability theory is to measure theory
KEine Ahnung Probability gives you information about future data/events based on a known underlying process. Statistics already has data/events from which an unknown underlying process can be discovered. That's why they're considered the inverse of each other since they start with what the other lacks and attempt to figure out what the other already has.
speaking of measure theory I went to the barber yesterday. he asked me what kind of haircut I wanted and I replied "give me the cantor set" I am now bald
I love explanations of how thinking tools we're invented. The dice-leading-to-probability story is gem for intellectual history, and a great example of a philosophical technology.
Yes! Probability is my favourite field of math! What I love about it is the implications of some of its results are amazing! Specially when applied to other fields of math, other sciences and even real life. For example, Central Limit Theorem and the Law of Big Numbers. Things that happen at random can have clear patterns with rational thinking and, why not, a bit intuition. (Though I guess all of my comment can be applied to Mathematics in general)
Egor Zvorykin Fair enough. Though I never claimed that Math could be successfully applied to everything (and if I did, it was unintentional), I totally agree that not all things can be abstracted in a mathematical sense. What I wanted to highlight, and apparently failed to do so, was that probability is not as fast in discarding intuition as other parts of math (for example, topology), and still maintaining the proper rigour in its proofs, leading to sometimes unexpected but yet interesting results. Then, that's only my personal opinion and perception.
The solution in the description is really curious, cause the probability for each one to win does not depend on how many points they need to get (I mean, it doesen't matter if they need to get 10 or 1000) as long as the points each of them need is the same; the probability each one has to win just depend on how many more points they need to win (It would be the same if they were 8 to 7 and they need to get 10, than if they were 98 to 97 and the goal was 100)
Probable, also means probe-able, testable, deterministic (which is generally taken as an antonym when meant totally), e.g. Can you probe a jar for cookies, depends on whether your hand fits its mouth, whether there are cookies in the jars, etc. This is mathematics: that statements are logical, and, arithmetical, and the consistency is left as homework...
Gerolamo Cardano who lived in the first half of the 16th century and known as the first to publish methods that solved reduced cubics also expressed probability by ratio
Its hard to say. different images of her show that freckle on her neck on different sides. the ones that seem more professional, match up with the freckles location on the vid so i would tepidly assume the vid is mirrored.
The thing I love about probability is that you can use it anywhere you aren't certain of the outcomes, you can determine what is most possible and what is not, also, I think this is why we use probabilistic models to describe the quantum world.
Since this video was about probability, and you mentioned that we want to make decisions rationally, I suggest doing a deeper exploration into this topic in a future video, and especially address Bayes' Theorem and Bayesian Reasoning. In particular, the host, Kelsey, may enjoy reading E.T. Jaynes' book "Probability Theory: The Logic of Science", which develops a framework for Bayesian reasoning from the ground up. I'm currently watching a video lecture series 'reading course' of Jaynes' book, which helps lower the slope of the learning curve of the book. Search TH-cam for 'Aubrey Clayton' or 'Probability Theory: The Logic of Science' to check it out. BTW: Jaynes, following R.T. Cox, develops the laws of probability *without* axioms, instead relying on an informal list of 'desiderata' which he formalizes in a general way, showing that Probability Theory is the *only* valid mathematical way to reason about 'plausibility'. This is an interesting alternative to Kolmogorov, and does not rely on Measure Theory.
I think the sort of numbers that engineers and physicists use, with a "margin of error", and "significant digits", imply a 1-dimensional interval, at least on the marginal part. The way that mathematicians use transcendental numbers, like "Real" decimals, seems analogous to the physics of wave-particle duality. A photon travels as a (higher dimensional) wave but interacts as a particle (at a 0-D point). Similarly a transcendental decimal transcends 0-dimensions and like a wave it can potentially intercept a range of points. Sure it may collapse to a single point, but this is a bit like the wave function collapsing. It's very interesting the way both of these examples cross dimensions, so to speak. It definitely serves a particular purpose in this way.
Probability is, IMHO, the most difficult of all mathematical subjects. This is because the topic appears, at first, to be completely intuitive but then very quickly, becomes unintuitive. It's torturous because one keeps trying to rely on intuition in understanding the concepts but it inevitably leads you down a blind alley. At least with algebra or analysis, the absence is easier grapple with because it never There it begin with. Not so probability!
Regarding ideas for future episodes, how about diving into Bachelier’s work on Mathematical Finance, Black-Scholes, etc? I know it’s heavy stuff but you guys manage to come up with such easy ways to explain really complex topics I’d be interested to see your take on that :) Thanks For the amazing work!
Did anyone else notice that the dice @6:10 are flawed? Opposite sides are supposed to add to 7 (1-6, 2-5, 3-4) but they have 5 and 6 opposite each other.
Bro why omit the origin of probability? The earliest known forms of probability and statistics were developed by Arab mathematicians studying cryptography between the 8th and 13th centuries. Most miracles originated in Levant.
Thank for all the good you do. I would like to learn more about 1. linear algebra (matrix calculations, algorithms, egen values, egen vector and so on) 2. fft or discrete fft (way and how it works, how to preserve amplitudes and so on) 3. approximating curves with sums of sin functions (how to kompres images using sums of sin) I know these are topics for engeners and I use some of them every day. But I only know them good enough to used dem. I want to have a better understanding of them. And I believe these are topics many people would benefit from know. Have a nice day.
pascal's wager is busted too easily, but then it is easy to explain it's interesting to see how you instructed the concept of the random variable, i still find it hard when i think about it and try and explain it to undergrad students in physics
I think of probability as a mathematical limit. Like, as the number of attempts N approaches infinity, the ratio of the number of times an event happens to N approaches the probability of the event occurring per attempt. Probably not mathematically correct, but that's just the way it seems to me.
For now my favourite part of probability theory is the existence of limit theorems, for series of indipendent or dependent variables. it's really curious to see how in the long run even random behaviors tend somehow to something stable. Plus stochastic process on their own are interesting for resolving PDE sometimes, as far as i know.
The fact that probability theory had a way of measuring, that covered both discrete and continuous, made me think of this as an analogy to general relativity and the quantum. If only physics had it so easy
A problem I've always had with Pascals Wager... A god is almost necessarily omnipotent. Omnipotence must convey the ability to violate causality and basic fundamental logic. In short, they can make paradoxes true in the most fundamental sense. This implies that first order logic (things like A is A and can not also be not-A at the same time in the same respect) is not universal and can not be relied upon. This requires abandoning all sense of an ability to reason effectively whatsoever. Showing that something leads to a logical contradiction is how we prove literally everything we believe to be false is false. Accepting the existence of an omnipotent being requires also accepting the possible or actual truth of all notions previously held as false.
Pascal's Wager has a lot of problems and there are many ways in which it is nonsense. One very obvious issue is that it doesn't account for the probability of the existence of other gods with other possibly contradictory conditions that you need to fulfill in order to earn the "reward" and in fact fulfilling one god's conditions may have earned you "punishment" by another one. Any one or even several out of the set of all gods that are imaginable may be the right one(s). The likelihood of picking the right one(s) and earning reward is zero since any finite number of gods has measure zero in the infinity of possibilities, but the likelihood of picking the wrong one and earning punishment is 100% since there is an infinity of wrong choices.
Dustin Rodriguez The only fundamental law on the universe is that change is the only constant, that means literaly anything can happen even if there is almost a zero chance.
@Juan Engazado I don't see what that has to do with what I said. If your system of determining truth permits fundamental logical paradoxes, things like "A is true. A is also not true." then you are incapable of determining the truth of literally anything. And in order to admit the possibility of an omnipotent being, you are required to accept those paradoxes, dismantling every single facet of understanding the world ever created. Neither formal systems of understanding nor intuitive systems based simply on a gut understanding of things can stand if you take away ALL ability to reason consistently as you must for an omnipotent entity to be able to exist. They must be able to create effects which have no cause. They must be able to do all of the things you allude to which do have zero chance of being possible. It is in their inherent nature. You don't have to believe they exist, but if you do its important to realize that it instantly becomes the only thing you can claim to believe. No other knowledge is possible in a universe where an omnipotent being exists.
Dustin Rodriguez Yeah i don't want to argue that much, however i find the implications of the classic problem of Godels (Second) Incompleteness theory is that or we may simply never know everything or our entire sets of rules of the universe cannot be proven with the rules of the universe itself which i don't agree so i stick with the first rule that if we will never know everything and something new is discovered every day for an infinite amount of time then everything that can happen or be discovered is possible so yeah its weird but existence itself is weird too so yeah i hope you understand me cause this topics are fun to discuss but i don't like really long discussions so take this as my last argument.
Is the probability of hitting a discrete point on the dart board zero, or a limit at zero? Are we just saying that the area of the point is zero, divide that by the total area (result still zero) and therefore calling it zero? I guess the apparent paradox arises from the fact that we're modeling the dart as impossibly thin?
The probability is a number. As such, it either is zero or isn't, we're not dealing with sequences/series, so limits don't really enter the question. The "probability of hitting a specific point", i.e. "the (uniform) measure of the set consisting of only that point", IS zero, no ifs and buts about it. As for how that arises... I don't really have an answer. In a sense, it's because there are so many points on the board, uncountably infinitely many, to be specific. There are a few similar paradoxes that appear when you're dealing with discrete points and the continuum simultaneously.
Well, series aren't completely out of the question when it comes to probabilities, it just depends on the case. In this particular situation they don't really enter the picture... And yeah, I remember having to think about the inifities/zeroes arising from looking at single points on e.g. number lines and such before. [EDIT: I mean if the dart board was shaped more weirdly and you couldn't determine the area with simple geometry, that'd be one instance where calculus immediately enters the picture]
The area of the point is exactly zero. The reason for this comes about by how we define "area" (specifically, Lebesgue measure). We define the area of a square to be s^2 where s is the side length. Also, we create the following rule: If an object A is encapsulated by object B, then the area of A is less than the the area of B. From these two assumptions, the area of a point *must* be 0. If we assume that the point has a nonzero area, then we could make a square smaller than this area which encapsulates the point. Our only option is to assign the point zero area.
A probability of zero does not mean something cannot happen. That is not because our model is not perfect or anything, but rather due to the fact that measures do not care for events that are possible, but so rare that betting on them repeatedly would lose you all your money no matter how good the odds or how much of your money you were betting. There is a similar notion to "probability of hitting a point" that works for points, though: the *density* (or density function) of a distibution, which is in a way the "derivative" of a distribution, since you can get the probability of hitting any area (no matter the shape) by integrating the density of the distribution in question over the area you want to know about (if it has a density: like derivatives of functions, a density does not always exist for a given distribution). When Kelsey talks about modifying the area of the dart board in such a way that you can still get the probability of hitting an area by calculating the area on the modified dart board, she is taking a similar, but different approach to the mathematicians who came up with densities: Not every distribution is uniform, so in general, certain areas are "worth more probability" than others. Kelsey just made the area that was "worth more" bigger until it was as big in proportion to its probability as every other area. The usual way of tackling this same problem is by defining a density function, that gives every point a certain weight. The probability of hitting that point is still zero, but if you integrate over that point, it will be worth more than a point where the density is lower. For example: the density of a uniform distribution is a constant function (well, duh, it's uniform, so every point has the same weight) on the area that can be hit, and either zero or non-defined anywhere else. The density of any discrete distribution is the same function as its probability function (since you don't run into the problem densities are supposed to solve in a discrete sample space). The density of a standard normal distribution is the function x |-> e^(-x^2) (a little "bump" around zero that tapers off towards higher values). The nice thing about the density is: Actual samples might be random, but if you have enough independent samples, they will start forming clusters around the areas that have highest density.
Yeah, I'm familiar with probability density functions and the like, but the notion of zero probability not being equivalent to impossible sounds weird to me. I could almost recall being told the opposite, though running through the math in my head it does seem to be the case. [EDIT: Then again, I don't think my uni course in probabilities and statistics utilized measure theory, so some simplifications might have been employed. I did skim a book on bayesian probability theory in the library once and it ran the same concepts by the reader in a more rigorous, measure theoretical way but I can't remember a whole lot of it (maybe the initial introduction of probability as measure/line segment or whatever) because all the jargon made my head spin and I only had an hour or so to look at it...]
Kevin Kim Note: it doesn't mean it's *not* impossible either - actually impossible things also have 0 measure. When talking about 0-measure-but-possible things, it's common to say it "almost never" happens. Same with measure 1 things that aren't guaranteed - they "almost always" happen. These phrases have specific meanings when probabilists talk. 😀
I don't agree with this claim. Saying something has zero probability actually DOES imply it's impossible! It's easy to see there's some funny-business going on here. Simply add-up all the probabilities of hitting any point. If these are all zero then the total probability of hitting any point is zero. So it's impossible to hit any point on the board. (My answer to this conundrum is in another thread above).
That "add up all the probabilities" part is the funny business here, but it's well-defined mathematical funny business. How many, exactly, is "all"? There's an *infinite* number of points. If each point has a non-zero probability, then when you "add them all up", you get a value of infinity! But probabilities have to sum to 1, so clearly there's something wrong in what we stated before - either there's not an infinite number of points, or the probability isn't non-zero. We know the first is true (if the points aren't infinite, you can tell me how many there are, and list them out; but I can always find another point that you missed), so it must have been the second assumption that's false - the probability of each point is indeed zero. This is no more paradoxical than asking what the "density" of the integers are in the rationals - it's also zero, even tho integers definitely exist in the rationals. There's just infinitely more rationals. ^_^
"Infinitesimals" don't exist in the reals. If you're using a number system with infinitesimals, then yeah, you're right - the probability of hitting a point on the dartboard is an infinitesimal. In systems without those, tho, it's just 0, and the math still works out fine. (Because in real limits, ∞×0 is an undefined form; it can evaluate to any number, and you can't tell which it is from the expression. You have to do some transform to get at the value some other way. In this particular case, the expression evaluates to 1.)
Continuous probability is a mathematical fantasy- useful for handling extremely short portions of possibility but really everything has a measure. A bad blind throwing dart player will not (if he hits the board at all) hit every point will equal probability (not that it really is the point of the exercise) depending on whether he is left/right handed and positioned relative to the board will introduce a consider bias to the placement of the darts- if LH then hitting more on the right and if below more on the higher part.
can u please make a video dealing with partial differential equations, I'm taking that course in a week so it might be nice if I could see an introduction by this incredible channel.
What I like about probability is the feeling after missing two 98% shots in a row after which your XCOM squad dies miserably. Also I like some angry comments of players who complain about the random numbers generator. That’s XCOM, that’s the life in our universe.
I concur with the claim that random variables are "coarse", though (9:55). With a little bit more work you could have defined a continuous bivariate distribution on the board that has the required measure and is as smooth as you like. Otherwise, great video! Keep up the great work!
If there are an infinite amount of exact points on the board, why doesn't that mean the probability of hitting one is infinitesimally small but not 0? Cause with the other examples like 1/4 the area fits on the board 4 times, with 1/2 it fits twice, so I was like if it fits on the board infinite amount of times it'd be 1/infinity? I'm just a layman so forgive me if I'm being stupid
The probability of hitting the board is 1, which is equal to the sum of all probabilities. If the probability of hitting an exact point is not zero, then the sum of all probabilities would become essentially infinitely large so it wouldn't work. See 3Blue1Brown's probability vids and his appearance on numberphile.
not a stupid question at all! In calculus, when you are taking a tiny element of length dx to integrate over, you take the limit as dx becomes infinitesimal. Contrast with this situation - a point can be APPROACHED by the limit of a vanishingly small element of area, but the point itself has 0 area, not an infinitesimal one. This next analogy might be wrong, but it's how I understand this: when integrating, you are "summing" COUNTABLY infinite length elements. With this dartboard, the area is a countable infinity of infinitesimal area elements, but it is an UNCOUNTABLE infinity of points. I hope this helps!!
My all time favorite application of probability theory is Denjoy's "proof' of the Riemann Hypothesis. Obviously it isn't a proof, otherwise the french mathematician Arnaud Denjoy would be much more famous. However, this probabilistic interpretation depends on (and fails because of) a simple assumption that intuitively seems to be correct: Consider the following (magical) coin that is flipped for every square-free positive integer n (that is there doesn't exist a square m^2 that divides n). If n has an even number of prime factors then the coin lands on heads; if n has an odd number of prime factors then the coin lands on tails. If we assume that this coin is a fair coin (that is the probability that it lands on heads/tails for any random square-free integer n is 50% and each flip is independent from the rest) then we can use the Central Limit Theorem in Probability to prove that the RH is true! This assumtion isn't completely unreasonable: why would any square-free integer have a higher probability of having an even number of prime factors than an odd number of prime factors. The problem with this assumption lies in the fact that each coin throw is not independent from previous throws. For more information check out Möbius' Arithmetical Function (this one calculates the parity in the number of prime factors), Merten's Function (partial sums of the series ∑µ(n) where µ is the Möbius function) and its implications for th RH, and last but not least Denjoy's Probabilistic Interpretation of the RH; an excelent description can be found in H.M. Edwards fascinating book "The Riemann Zeta Function" (pg 268) P.S. I actually have both pages of Edward's Book where Denjoy's Interpretation appears (sadly I don't have the full book), so if anyone is interested here's the link: drive.google.com/file/d/0B6FUVGLkvRJbYXRMNndMbC1JdDQ/view?usp=sharing
I have just come across your videos, and find them very informative. i wonder if you have ever done a video on what is sometimes called "The Monty Hall Game Show problem" Thanks for the excellent video.
I was looking for something with more history to be fair. Especially the axiomatic approach of Kolmogorov's vs the limit approach for probability, and early origins of what happened afterwards.
The fundamental principle of probability is that as time goes by the probability does not change. If I have spun 5 heads in a row, we all shout that the next spin is 50/50, year 7 shout out. So why is it that when one studies those who are banned from casinos the common thread is 'missing outcomes'.
There are many interesting things to say about probability. An example: very often "normal" probability distributions are used in statistical approaches to answer questions that are not in fact normally distributed in real life. Think e.g. of assessing the risk of a plane losing its wings due to a vertical gust of air or any other risk assessment. I understand mathematicians don't want to get their hands dirty with reality, but it would be a lot more interesting if you did imho.
Isn't the circle formed by many points? If so the some of all the points of an area should vê equal to it's area, so the some of the probability of multiplus points of an area should be equal to the probability of the dard being in some point of an area. Like, we can say that a patch of 6 cookies is made of 1 cookie+ 1 +1+1+1+1, the somatory of the parts is equal to the total
Using units as a descriptor for a number is a good way to not lose the context of what the number was supposed to represent. Units can be used just like equations, in what is called "dimensional analysis". For consistency the units ought to match the equations. One result can be compared to the other as a way of "checking your work".
Imagine you would have to guess a number. This number can be any number you can think of (0, 1, 2, 3, 4, 5...). Since there are an invinite amount of numbers, the chance of winning is 1/infinite. Does that make the challenge impossible?
I someone is interested of the actual axioms of a measure: A measure is a functiondefined on a certain type (which is a little bit more complicated to explain) of subsets of a specifc set X It satisfies: The measure of every set is a positive real number or 0 The measure of nothing is 0 The measure of a countable union of sets which have no intersection is the sum of the measures of the sets
Haven't heard it either. People do call 3 dimensional objects squares, like a square of cardboard, but that seems to be just because they approximate 2 dimensional objects.
I think pseudo-random number generators are interesting too. Have people studied the probability of those generators actually producing "random" numbers?
suppose you are playing a game in which you win $15 if heads come or lose $10 if tails come. if you toss the coin 10 times then you might not win anything or lose the money but as you toss the coin many times (say 100) then the result will get closer the probabilistic value i.e 50/100=1/2 and you will win money so in a bet which gives you advantage it is advisable to play maximum number of times possible
I wouldn't honestly place Pascal's Wager in the probability discussion, as he was attempting more to use rational thinking about probability and applying it to theological thought with a particular bias in the first place.
Pascal's wager is logically flawed. It only works if there is only one possible god that can be imagined. But in fact quite the opposite is the case, there is an infinity of possible imaginable gods. So any probabilistic reasoning has to take all of them into account and many are mutually exclusive - what earns you reward with one may earn you punishment with another.
I'm still not convinced of the solution (or the logic) of that famous problem: "You choose between 3 doors, one has a price. After you choose someone opens a door without a price among the ones you didnt choose. You then have the choice of changing your door to the remaining door. Is it wise to change it?" So, the answer is "yes" because of the probability of the first choice "1/3 chance of getting the winner door" and "2/3 chance of the winner door being among the others". However, my question would be: Since theres 100% chance that there is a empty door among the ones no choosen, and the revealed door is always an empty one (it is not a random choice), why is the probability of the winner door added to the non-choosed door instead of removing it from the spectrum? In the end, I have a door and theres another door, one is a winner. shouldnt it be a 1/2 chance?
+Francisco Roy De Mare You can't ignore the history of the scenario. If I ask you what the probability is that the top card in a 52 card standard deck is the 3 of clubs, the answer is obviously 1/52. But if I show you the bottom card in that deck and it is the 3 of clubs, then ask you what the probability is that the top card is the 3 of clubs, the probability is clearly 0. In both cases you have a 52 card deck, but the fact that the current state is the same in both situations is irrelevant. You have 2 doors, but those 2 doors took different paths to get where they are. The door you chose has a 1/3 chance of having the prize behind it. It had a 1/3 chance when you chose it and, as far as you know, nothing has happened to it or the prize since then. The 2 doors you didn't choose had a 2/3 chance of having the prize behind one of them. That too has not changed. The 2 doors you didn't choose still have a 2/3 chance of having the prize behind one of them. But, because you know for certain that the prize is not behind the door that was opened, the entire 2/3 chance of having the prize is with the other door.
Yes, I understand that. And probably a sample test (repeat this scenario 10000 times would confirm), but I still cant see the objective logical demostration. Your card example makes no sense, its like hes showing me that one the other doors has the price. Like I said, the revelead door is ALWAYS empty, its not random. So theres only 2 doors in the game, one with price and one without.
Its very counter intuitive, thats why the problem is famous (?) My problem is that if I was watching the show the previous day, my first choice of door is irrelevant, whatever I choose the winning door or a empty door, Im still picking between two at the end. So "Is it wise to change it?" makes no sense for me, 2 doors, 1 choice... maybe "Is it more likely you get the right door twice?" 3 doors, 2 choices.
I completely missed this when it first appeared due to what I think was insanity. I'm not sure the insanity is over but I just wanted to say that I think there is a hidden secret in coin flipping. It might even be tied to universe being a simulation. But remember, I'm not sure the insanity is over.
A next-door-neighbour subject is that of reasoning given only probabilities and events. That is, given that event A has happened, and a bunch of probabilities, what is the probability of event B?
Bayes theorem arises naturally (ot, to be more precise, it is a natural definition for the intuitive notion of a conditional probability). Assume you have a probability measure space X (meaning you defined a measure/"size" P for subsets of X and X has measure/probability 1, ergo P(x)=1) and a subset A of X. Then you natually also get a a new measure for subsets B of X by taking P(intersection of A and B). This new measure has one problem though: The new measure of X is no longer 1, but the smaller P(intersection of A and X)=P(A). To once again get a probability of 1 for X, you need to "normalize" the measure by the factor P(A). So in total the new measure then reads "new measure of B" is "P(intersection of A and B)/P(A)". This new measure is called the probability of B under the condition A. As a stochastic explanation: A and B are the events (as subsets of the set of all results). You know know that A already happened, so you only need to consider points in A and want to know the measure of points in B. So you take set of all the points who lie in both A and B the intersection). But because you want to model the fact that A already happened, A must have probability 1, so you need to normalize the whole thing
Could it not have been based on measure theory? Would a different initial aproach make certain answers easier to reach? Would it make it apply more directly to quantum computing?
When you start talking about the dart board and hitting a single point on that dart board: you need to properly define that point. If you expressly mean hitting exactly a 0-dimensional point and no other points around it, then that is impossible...but it's also somewhat redundant to mention. If however, you mean that each point is infinitesimal in size (and this is typically what people do mean, as they are dividing the board into an infinite number of sections), and that the dart head is of a comparable size (i.e. also infinitesimal as already stated in the video), then the probability of hitting any specific point becomes the proportion of that point to the overall size of the board which is infinitesimal and clearly non-zero. Applying a real number field to this type of maths where you either take limits or involve certain types of infinities/infinitesimals is erroneous. Instead you should be applying a number field that includes these things as numbers (e.g. the surreal number field), which would lead to much less confusion over such things as inaccurate approximations are no longer used.
Practically when you hit anywhere on dart board, you are simultaneously hitting a Planck area of points. So the probability of hitting any given point must be 1/(No. of plank Areas on the dart board) which is non zero. Just saying...
How are you defining Planck area? You seem to treating Planck area as an object, rather than a unit of measure. If Planck area is a unit of measure, each dart lands in uncountably many circular disks each with a area of 1 Planck area.
Why? If the probability that the dart will hit point A is (Area of 1 Plank Area)/(Area of the dart board), then hitting A must mean hitting the unique subset of the dartboard with area 1 Planck that contains A But there is no such subset. Uncountably many subsets of the dartboard have area 1 Planck and contain A. You could select a finite subset of all the mathematical points on the board and designate the disks centered on those points as physical points. (Mathematical points in overlapping disks can be assigned to one or the other somehow.) If you chose wisely, the finite set of k disks covers the board and the probability of hitting a point could be said to be 1/k. k would always be larger than (Area of the dart board)/(Area of 1 Plank Area) due to overlap. There are uncountably many finite subsets of the set of points on the dartboard. For any of those, there is always one that makes the probability distribution more uniform. (That's based on my method of assigning mathematical points to physical ones in ambiguous cases. There might be an assignment method that does not skew the distribution.) Do we have any reason to believe that any one set is the right one?
My logic is when the dart is actually hitting any point within the Planck area circle around point A, it is also simultaneously hitting point A. Any measurement system will give same response for the events of dart hitting Point A & dart hitting some other point within the Planck area circle around Point A. So dart hitting any Point within the Planck area circle around point A should also be considered as hitting Point A.
Let r be the radius of a circle with an area equal to 1 Planck area. Suppose point B is hit by the dart. The dart is treated as if it hit point A, a point within r of B. But uncountably many points are within r of B. Which one is point A?
+You Tube Two points. First, many scientists and mathematicians held and hold absurd beliefs. Second, I think Pascal only said the loss was small in comparison to the possible gain. I think if Pascal had understood measure theory he might have reached a different conclusion. He assumed without proof that, in the absence of an afterlife, his lifespan had finite measure.
Well he's either laughing at you or doesn't care, you on the other hand look stupid in both cases. Scientists are just people, although science pretends to be an absolute truth it'll always be just a concept
Very interesting the kind how to treat probabillity as measure. I would like to launch a problem in the channel about measure. If I divid a pizza in equal pieces, the number dosen't matter but to make it easy, in 8 equal parts, wich part is the biggest? I call it the paradox of the stretch' time.
Isn't the probability of hitting any point on the dart board actually 1/infinity ? Seems like asserting that the probability is 0 is the same as asserting that 1/infinity = 0
that's because any finite number divided by any infinity is equal to zero. At first glance it might seem paradoxical until you realize we are talking about a freakin' infinity....
+Graham Henry, I agree. As I understand it, limit functions are needed to define "infinity" , and to define or evaluate "1/infinity". But this seems to get lost to the confusing vagaries of so called "Real Numbers". It's perhaps more clear to stay within a 2-Dimensional space based on areas, and then apply limits if necessary.
Physically there's a finite number of points with each atom on the surface of the board representing a point. So there's 1/# of atoms. And if you want to be practical and not as precise, just split the board into hundreds of equal parts and throw the dart. Now the probability is 1/# of equal parts, which is more useful
I've always thought probabillity isn't real. If we look at the darts example, we can (in theory) accurately calculate where a dart will hit if we take into consideration the relevent data (Player's strength, dart's direction and weight distribution, wind, coriolis effect, etc). I do however find probability theory convenient for engineering.
Quantum mechanics, which determines the behavior of the tiny particles that make up everything, is fundamentally probabilistic. Wave functions and particle interactions in quantum mechanics and quantum field theory are most accurately described as a set of probabilities, and are capable of describing nature more accurately than classical physics. And although the randomness diminishes as the scale gets bigger, it's still there. In other words, there is a 99.99999999999999999999999999999999999999999999999999999999999999999999999999999% (perhaps even higher) chance that the dart will land in the spot predicted by factoring in every aspect of the air around it, the way it was thrown, the mass distribution of the dart, and the gravity of every last particle around the dart, but you cannot say with complete absolute certainty that it will land in the spot you think it will. Also, Heisenberg's uncertainty principle says that you cannot know the position or the momentum of a particle with exact certainty, and the more accurately you know its momentum the less accurately you can know its position. These effects diminish at larger scales because the relation between these uncertainties only becomes relevant at the atomic scale. So probability isn't real in an absolute sense when dealing with classical physics, which is mostly relevant on the scale of darts and scoreboards, but in the complete picture of the fundamental workings of the universe, probability is "real" in the sense that you are talking about as there are things that can only be described (or are most accurately described) as probabilities.
the area of the dart is assumed infinitally small ..not zero...so probability of hitting the board at a point is not zero at all....whats the paradoxical about that.?
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THANK YOU for discussing probability WITHOUT mentioning statistics!
Probability and statistics are often lumped together as if they formed one big word. Addressing each separately is necessary before we can realize the immense power inherent in using them together.
BobC YES. I'm guessing another fellow probabilist. 🙌
They are very closely related though. I mean statistics is basically applied probability. You use the tools developed in probability theory almost exclusively. And probability theory without statistics would be half as interesting, as you wouldn't have an application of it.
But I guess mathematics is also closely related to physics and IT in that sense, so I see where you are coming from.
I wonder though if that episode was understandable without already knowing about probability theory. The explanation of random variables for example was extremely short. I doubt anyone could understand random variables from that short comment.
Statistics is usually a tool for constructing a probability sample. Hail the dice!
In what sense is statistic the inverse of probability? That would imply that they cancel out in some sense. Which doesn't make any sense so it seems like you just threw words around.
While statistics is heavily dependent on probability theory. You use the models and axioms of probability theory like Random variables, stochastic processes, etc. And you use the theorems of probability theory like the different central limit theorems, law of large numbers, Bayes, measurable mappings of independent random variables are still independent, etc. to base all your theorems of statistic on. Or you can justify using the central limit theorem that you assume something is normal distributed. Everything in statistics is based on probability theory. You could probably say statistics is to probability theory, what probability theory is to measure theory
KEine Ahnung Probability gives you information about future data/events based on a known underlying process. Statistics already has data/events from which an unknown underlying process can be discovered. That's why they're considered the inverse of each other since they start with what the other lacks and attempt to figure out what the other already has.
*Me thinking*
Yes, summer vacations, no school for weeks!
Now let's watch a video about mathmatics...
MineWarz imagine that, but going to math camps and competitions during the summer. that's me
So there are people who go to those math camps, interesting...
MineWarz well, they wouldn't exist if nobody went to them
relevancy 100
Guess who’s doing that 4 years later
speaking of measure theory I went to the barber yesterday.
he asked me what kind of haircut I wanted and I replied "give me the cantor set"
I am now bald
+Daniel Flores You only look bald because the probability of a photon bouncing off your hair has probability 0.
You are bald because you only have finitely many hair follicles
Should have asked for a fat cantor set.
I should still have an infinitely uncountable number of hairs if i have the cantor set as a haircut
+Daniel Flores So the barber added hairs.
I love explanations of how thinking tools we're invented. The dice-leading-to-probability story is gem for intellectual history, and a great example of a philosophical technology.
Yes! Probability is my favourite field of math!
What I love about it is the implications of some of its results are amazing! Specially when applied to other fields of math, other sciences and even real life.
For example, Central Limit Theorem and the Law of Big Numbers. Things that happen at random can have clear patterns with rational thinking and, why not, a bit intuition.
(Though I guess all of my comment can be applied to Mathematics in general)
It is important to remember when applying any math to real world that it's just a concept, meaning it could easily be not compatible with what we see.
Egor Zvorykin Fair enough. Though I never claimed that Math could be successfully applied to everything (and if I did, it was unintentional), I totally agree that not all things can be abstracted in a mathematical sense.
What I wanted to highlight, and apparently failed to do so, was that probability is not as fast in discarding intuition as other parts of math (for example, topology), and still maintaining the proper rigour in its proofs, leading to sometimes unexpected but yet interesting results.
Then, that's only my personal opinion and perception.
Your take on Gödel's incompleteness theorems would be interesting.
The solution in the description is really curious, cause the probability for each one to win does not depend on how many points they need to get (I mean, it doesen't matter if they need to get 10 or 1000) as long as the points each of them need is the same; the probability each one has to win just depend on how many more points they need to win
(It would be the same if they were 8 to 7 and they need to get 10, than if they were 98 to 97 and the goal was 100)
Poor Pascal. He must have been unaware of other religions around the world. Lol.
Probable, also means probe-able, testable, deterministic (which is generally taken as an antonym when meant totally), e.g. Can you probe a jar for cookies, depends on whether your hand fits its mouth, whether there are cookies in the jars, etc. This is mathematics: that statements are logical, and, arithmetical, and the consistency is left as homework...
Gerolamo Cardano who lived in the first half of the 16th century and known as the first to publish methods that solved reduced cubics also expressed probability by ratio
What is the probability that Kelsey is wearing that wedding ring to diminish the probability of random internet math nerds proposing?
Do you mean that the video is mirrored? The ring appears to be in her right hand.
Its hard to say. different images of her show that freckle on her neck on different sides. the ones that seem more professional, match up with the freckles location on the vid so i would tepidly assume the vid is mirrored.
The thing I love about probability is that you can use it anywhere you aren't certain of the outcomes, you can determine what is most possible and what is not, also, I think this is why we use probabilistic models to describe the quantum world.
Since this video was about probability, and you mentioned that we want to make decisions rationally, I suggest doing a deeper exploration into this topic in a future video, and especially address Bayes' Theorem and Bayesian Reasoning.
In particular, the host, Kelsey, may enjoy reading E.T. Jaynes' book "Probability Theory: The Logic of Science", which develops a framework for Bayesian reasoning from the ground up. I'm currently watching a video lecture series 'reading course' of Jaynes' book, which helps lower the slope of the learning curve of the book. Search TH-cam for 'Aubrey Clayton' or 'Probability Theory: The Logic of Science' to check it out. BTW: Jaynes, following R.T. Cox, develops the laws of probability *without* axioms, instead relying on an informal list of 'desiderata' which he formalizes in a general way, showing that Probability Theory is the *only* valid mathematical way to reason about 'plausibility'. This is an interesting alternative to Kolmogorov, and does not rely on Measure Theory.
I think the sort of numbers that engineers and physicists use, with a "margin of error", and "significant digits", imply a 1-dimensional interval, at least on the marginal part.
The way that mathematicians use transcendental numbers, like "Real" decimals, seems analogous to the physics of wave-particle duality. A photon travels as a (higher dimensional) wave but interacts as a particle (at a 0-D point). Similarly a transcendental decimal transcends 0-dimensions and like a wave it can potentially intercept a range of points. Sure it may collapse to a single point, but this is a bit like the wave function collapsing. It's very interesting the way both of these examples cross dimensions, so to speak. It definitely serves a particular purpose in this way.
This video should be shown at the first class of every probability course ...
Probability is, IMHO, the most difficult of all mathematical subjects. This is because the topic appears, at first, to be completely intuitive but then very quickly, becomes unintuitive. It's torturous because one keeps trying to rely on intuition in understanding the concepts but it inevitably leads you down a blind alley. At least with algebra or analysis, the absence is easier grapple with because it never There it begin with. Not so probability!
Joshua Sherwin probably
Regarding ideas for future episodes, how about diving into Bachelier’s work on Mathematical Finance, Black-Scholes, etc? I know it’s heavy stuff but you guys manage to come up with such easy ways to explain really complex topics I’d be interested to see your take on that :) Thanks For the amazing work!
Did anyone else notice that the dice @6:10 are flawed? Opposite sides are supposed to add to 7 (1-6, 2-5, 3-4) but they have 5 and 6 opposite each other.
Bro why omit the origin of probability?
The earliest known forms of probability and statistics were developed by Arab mathematicians studying cryptography between the 8th and 13th centuries.
Most miracles originated in Levant.
Thank for all the good you do.
I would like to learn more about
1. linear algebra (matrix calculations, algorithms, egen values, egen vector and so on)
2. fft or discrete fft (way and how it works, how to preserve amplitudes and so on)
3. approximating curves with sums of sin functions (how to kompres images using sums of sin)
I know these are topics for engeners and I use some of them every day. But I only know them good enough to used dem. I want to have a better understanding of them. And I believe these are topics many people would benefit from know.
Have a nice day.
You should make a video on Bayes' Theorem. It is crucial for the understanding of the scientific method.
My favourite aspect of probability is knowing how often I'll hit a straight or flush draw when the other players at the table have no clue
pascal's wager is busted too easily, but then it is easy to explain
it's interesting to see how you instructed the concept of the random variable, i still find it hard when i think about it and try and explain it to undergrad students in physics
More interesting, if she were to describe the constant variable.
I think of probability as a mathematical limit. Like, as the number of attempts N approaches infinity, the ratio of the number of times an event happens to N approaches the probability of the event occurring per attempt.
Probably not mathematically correct, but that's just the way it seems to me.
For now my favourite part of probability theory is the existence of limit theorems, for series of indipendent or dependent variables. it's really curious to see how in the long run even random behaviors tend somehow to something stable. Plus stochastic process on their own are interesting for resolving PDE sometimes, as far as i know.
The fact that probability theory had a way of measuring, that covered both discrete and continuous, made me think of this as an analogy to general relativity and the quantum. If only physics had it so easy
A problem I've always had with Pascals Wager... A god is almost necessarily omnipotent. Omnipotence must convey the ability to violate causality and basic fundamental logic. In short, they can make paradoxes true in the most fundamental sense. This implies that first order logic (things like A is A and can not also be not-A at the same time in the same respect) is not universal and can not be relied upon. This requires abandoning all sense of an ability to reason effectively whatsoever. Showing that something leads to a logical contradiction is how we prove literally everything we believe to be false is false. Accepting the existence of an omnipotent being requires also accepting the possible or actual truth of all notions previously held as false.
Pascal's Wager has a lot of problems and there are many ways in which it is nonsense.
One very obvious issue is that it doesn't account for the probability of the existence of other gods with other possibly contradictory conditions that you need to fulfill in order to earn the "reward" and in fact fulfilling one god's conditions may have earned you "punishment" by another one. Any one or even several out of the set of all gods that are imaginable may be the right one(s). The likelihood of picking the right one(s) and earning reward is zero since any finite number of gods has measure zero in the infinity of possibilities, but the likelihood of picking the wrong one and earning punishment is 100% since there is an infinity of wrong choices.
Dustin Rodriguez The only fundamental law on the universe is that change is the only constant, that means literaly anything can happen even if there is almost a zero chance.
@Juan Engazado I don't see what that has to do with what I said. If your system of determining truth permits fundamental logical paradoxes, things like "A is true. A is also not true." then you are incapable of determining the truth of literally anything. And in order to admit the possibility of an omnipotent being, you are required to accept those paradoxes, dismantling every single facet of understanding the world ever created. Neither formal systems of understanding nor intuitive systems based simply on a gut understanding of things can stand if you take away ALL ability to reason consistently as you must for an omnipotent entity to be able to exist. They must be able to create effects which have no cause. They must be able to do all of the things you allude to which do have zero chance of being possible. It is in their inherent nature. You don't have to believe they exist, but if you do its important to realize that it instantly becomes the only thing you can claim to believe. No other knowledge is possible in a universe where an omnipotent being exists.
Dustin Rodriguez Yeah i don't want to argue that much, however i find the implications of the classic problem of Godels (Second) Incompleteness theory is that or we may simply never know everything or our entire sets of rules of the universe cannot be proven with the rules of the universe itself which i don't agree so i stick with the first rule that if we will never know everything and something new is discovered every day for an infinite amount of time then everything that can happen or be discovered is possible so yeah its weird but existence itself is weird too so yeah i hope you understand me cause this topics are fun to discuss but i don't like really long discussions so take this as my last argument.
Is the probability of hitting a discrete point on the dart board zero, or a limit at zero? Are we just saying that the area of the point is zero, divide that by the total area (result still zero) and therefore calling it zero? I guess the apparent paradox arises from the fact that we're modeling the dart as impossibly thin?
The probability is a number. As such, it either is zero or isn't, we're not dealing with sequences/series, so limits don't really enter the question.
The "probability of hitting a specific point", i.e. "the (uniform) measure of the set consisting of only that point", IS zero, no ifs and buts about it.
As for how that arises... I don't really have an answer. In a sense, it's because there are so many points on the board, uncountably infinitely many, to be specific. There are a few similar paradoxes that appear when you're dealing with discrete points and the continuum simultaneously.
Well, series aren't completely out of the question when it comes to probabilities, it just depends on the case. In this particular situation they don't really enter the picture...
And yeah, I remember having to think about the inifities/zeroes arising from looking at single points on e.g. number lines and such before.
[EDIT: I mean if the dart board was shaped more weirdly and you couldn't determine the area with simple geometry, that'd be one instance where calculus immediately enters the picture]
The area of the point is exactly zero. The reason for this comes about by how we define "area" (specifically, Lebesgue measure). We define the area of a square to be s^2 where s is the side length. Also, we create the following rule: If an object A is encapsulated by object B, then the area of A is less than the the area of B. From these two assumptions, the area of a point *must* be 0. If we assume that the point has a nonzero area, then we could make a square smaller than this area which encapsulates the point. Our only option is to assign the point zero area.
A probability of zero does not mean something cannot happen. That is not because our model is not perfect or anything, but rather due to the fact that measures do not care for events that are possible, but so rare that betting on them repeatedly would lose you all your money no matter how good the odds or how much of your money you were betting.
There is a similar notion to "probability of hitting a point" that works for points, though: the *density* (or density function) of a distibution, which is in a way the "derivative" of a distribution, since you can get the probability of hitting any area (no matter the shape) by integrating the density of the distribution in question over the area you want to know about (if it has a density: like derivatives of functions, a density does not always exist for a given distribution).
When Kelsey talks about modifying the area of the dart board in such a way that you can still get the probability of hitting an area by calculating the area on the modified dart board, she is taking a similar, but different approach to the mathematicians who came up with densities: Not every distribution is uniform, so in general, certain areas are "worth more probability" than others. Kelsey just made the area that was "worth more" bigger until it was as big in proportion to its probability as every other area. The usual way of tackling this same problem is by defining a density function, that gives every point a certain weight. The probability of hitting that point is still zero, but if you integrate over that point, it will be worth more than a point where the density is lower.
For example: the density of a uniform distribution is a constant function (well, duh, it's uniform, so every point has the same weight) on the area that can be hit, and either zero or non-defined anywhere else.
The density of any discrete distribution is the same function as its probability function (since you don't run into the problem densities are supposed to solve in a discrete sample space).
The density of a standard normal distribution is the function x |-> e^(-x^2) (a little "bump" around zero that tapers off towards higher values).
The nice thing about the density is: Actual samples might be random, but if you have enough independent samples, they will start forming clusters around the areas that have highest density.
Yeah, I'm familiar with probability density functions and the like, but the notion of zero probability not being equivalent to impossible sounds weird to me. I could almost recall being told the opposite, though running through the math in my head it does seem to be the case.
[EDIT: Then again, I don't think my uni course in probabilities and statistics utilized measure theory, so some simplifications might have been employed. I did skim a book on bayesian probability theory in the library once and it ran the same concepts by the reader in a more rigorous, measure theoretical way but I can't remember a whole lot of it (maybe the initial introduction of probability as measure/line segment or whatever) because all the jargon made my head spin and I only had an hour or so to look at it...]
The Riemann Hypothesis could be a great subject for here. Seen some great videos about it, think you can do it better ^^
"Saying something has zero probability does not imply it's impossible."
Mind. Blown.
(Oh, and you forgot "town square"!)
Kevin Kim Note: it doesn't mean it's *not* impossible either - actually impossible things also have 0 measure. When talking about 0-measure-but-possible things, it's common to say it "almost never" happens. Same with measure 1 things that aren't guaranteed - they "almost always" happen. These phrases have specific meanings when probabilists talk. 😀
I don't agree with this claim. Saying something has zero probability actually DOES imply it's impossible! It's easy to see there's some funny-business going on here. Simply add-up all the probabilities of hitting any point. If these are all zero then the total probability of hitting any point is zero. So it's impossible to hit any point on the board. (My answer to this conundrum is in another thread above).
That "add up all the probabilities" part is the funny business here, but it's well-defined mathematical funny business. How many, exactly, is "all"? There's an *infinite* number of points. If each point has a non-zero probability, then when you "add them all up", you get a value of infinity! But probabilities have to sum to 1, so clearly there's something wrong in what we stated before - either there's not an infinite number of points, or the probability isn't non-zero. We know the first is true (if the points aren't infinite, you can tell me how many there are, and list them out; but I can always find another point that you missed), so it must have been the second assumption that's false - the probability of each point is indeed zero.
This is no more paradoxical than asking what the "density" of the integers are in the rationals - it's also zero, even tho integers definitely exist in the rationals. There's just infinitely more rationals. ^_^
>If each point has a non-zero probability, then when you "add them all up", you get a value of infinity!
Not if it's "infinitesimal" probability.
"Infinitesimals" don't exist in the reals. If you're using a number system with infinitesimals, then yeah, you're right - the probability of hitting a point on the dartboard is an infinitesimal. In systems without those, tho, it's just 0, and the math still works out fine. (Because in real limits, ∞×0 is an undefined form; it can evaluate to any number, and you can't tell which it is from the expression. You have to do some transform to get at the value some other way. In this particular case, the expression evaluates to 1.)
9:22 you just gave a depiction of how the cortex learns a skill (except it does have a lovely reversable characteristic too it)
Continuous probability is a mathematical fantasy- useful for handling extremely short portions of possibility but really everything has a measure. A bad blind throwing dart player will not (if he hits the board at all) hit every point will equal probability (not that it really is the point of the exercise) depending on whether he is left/right handed and positioned relative to the board will introduce a consider bias to the placement of the darts- if LH then hitting more on the right and if below more on the higher part.
Bayes Theorem is my favourite.
7:13 the point (1/2 , 1/2) does not lie on a circle with radius 1/sqrt(pi) so it actually is impossible.
can u please make a video dealing with partial differential equations, I'm taking that course in a week so it might be nice if I could see an introduction by this incredible channel.
Love how, at 4:13 neither of the pictures shows Algebra or Geometry (Geometry that deals with constructions, that is)
What I like about probability is the feeling after missing two 98% shots in a row after which your XCOM squad dies miserably. Also I like some angry comments of players who complain about the random numbers generator. That’s XCOM, that’s the life in our universe.
I used to play XCOM.
I concur with the claim that random variables are "coarse", though (9:55). With a little bit more work you could have defined a continuous bivariate distribution on the board that has the required measure and is as smooth as you like. Otherwise, great video! Keep up the great work!
Favorite? It has to be the Monty Hall problem! It even bugged Paul Erdos, who was incredulous until running a computer simulation...
If there are an infinite amount of exact points on the board, why doesn't that mean the probability of hitting one is infinitesimally small but not 0? Cause with the other examples like 1/4 the area fits on the board 4 times, with 1/2 it fits twice, so I was like if it fits on the board infinite amount of times it'd be 1/infinity? I'm just a layman so forgive me if I'm being stupid
The probability of hitting the board is 1, which is equal to the sum of all probabilities. If the probability of hitting an exact point is not zero, then the sum of all probabilities would become essentially infinitely large so it wouldn't work. See 3Blue1Brown's probability vids and his appearance on numberphile.
not a stupid question at all! In calculus, when you are taking a tiny element of length dx to integrate over, you take the limit as dx becomes infinitesimal. Contrast with this situation - a point can be APPROACHED by the limit of a vanishingly small element of area, but the point itself has 0 area, not an infinitesimal one. This next analogy might be wrong, but it's how I understand this: when integrating, you are "summing" COUNTABLY infinite length elements. With this dartboard, the area is a countable infinity of infinitesimal area elements, but it is an UNCOUNTABLE infinity of points.
I hope this helps!!
@@ronshvartsman7630 Thanks man :) I'd just accepted it at this point but you bring up a nice way to think about it
My all time favorite application of probability theory is Denjoy's "proof' of the Riemann Hypothesis. Obviously it isn't a proof, otherwise the french mathematician Arnaud Denjoy would be much more famous. However, this probabilistic interpretation depends on (and fails because of) a simple assumption that intuitively seems to be correct:
Consider the following (magical) coin that is flipped for every square-free positive integer n (that is there doesn't exist a square m^2 that divides n). If n has an even number of prime factors then the coin lands on heads; if n has an odd number of prime factors then the coin lands on tails. If we assume that this coin is a fair coin (that is the probability that it lands on heads/tails for any random square-free integer n is 50% and each flip is independent from the rest) then we can use the Central Limit Theorem in Probability to prove that the RH is true! This assumtion isn't completely unreasonable: why would any square-free integer have a higher probability of having an even number of prime factors than an odd number of prime factors. The problem with this assumption lies in the fact that each coin throw is not independent from previous throws.
For more information check out Möbius' Arithmetical Function (this one calculates the parity in the number of prime factors), Merten's Function (partial sums of the series ∑µ(n) where µ is the Möbius function) and its implications for th RH, and last but not least Denjoy's Probabilistic Interpretation of the RH; an excelent description can be found in H.M. Edwards fascinating book "The Riemann Zeta Function" (pg 268)
P.S. I actually have both pages of Edward's Book where Denjoy's Interpretation appears (sadly I don't have the full book), so if anyone is interested here's the link: drive.google.com/file/d/0B6FUVGLkvRJbYXRMNndMbC1JdDQ/view?usp=sharing
I have just come across your videos, and find them very informative. i wonder if you have ever done a video on what is sometimes called "The Monty Hall Game Show problem" Thanks for the excellent video.
I was looking for something with more history to be fair. Especially the axiomatic approach of Kolmogorov's vs the limit approach for probability, and early origins of what happened afterwards.
The fundamental principle of probability is that as time goes by the probability does not change. If I have spun 5 heads in a row, we all shout that the next spin is 50/50, year 7 shout out. So why is it that when one studies those who are banned from casinos the common thread is 'missing outcomes'.
I feel bad for the person who posed so happily to be called as "a boring and old fashioned person" XD
There are many interesting things to say about probability. An example: very often "normal" probability distributions are used in statistical approaches to answer questions that are not in fact normally distributed in real life. Think e.g. of assessing the risk of a plane losing its wings due to a vertical gust of air or any other risk assessment. I understand mathematicians don't want to get their hands dirty with reality, but it would be a lot more interesting if you did imho.
Tonight on a very special pbs infinite, a troubled gambler reaches out to a mathematician for help.
Isn't the circle formed by many points? If so the some of all the points of an area should vê equal to it's area, so the some of the probability of multiplus points of an area should be equal to the probability of the dard being in some point of an area.
Like, we can say that a patch of 6 cookies is made of 1 cookie+ 1 +1+1+1+1, the somatory of the parts is equal to the total
Using units as a descriptor for a number is a good way to not lose the context of what the number was supposed to represent. Units can be used just like equations, in what is called "dimensional analysis". For consistency the units ought to match the equations. One result can be compared to the other as a way of "checking your work".
Hello PBS, where can I find the music you use for Infinite Series (and Spacetime) videos please? Thanks.
Bruh...
PBS is pushing TH-cam that I can see a 'Latex' version and 'join discussion' in the comments soon.
Imagine you would have to guess a number. This number can be any number you can think of (0, 1, 2, 3, 4, 5...). Since there are an invinite amount of numbers, the chance of winning is 1/infinite. Does that make the challenge impossible?
Please reduce the volume of PBS logo intro...
lmao Didnt expect this comment coz LIKE LITERALLY 5 SECONDS AGO IT HAPPENED LOUD
I someone is interested of the actual axioms of a measure:
A measure is a functiondefined on a certain type (which is a little bit more complicated to explain) of subsets of a specifc set X
It satisfies:
The measure of every set is a positive real number or 0
The measure of nothing is 0
The measure of a countable union of sets which have no intersection is the sum of the measures of the sets
You should mention that this certain type is Sigma algebra
8:38 That looks like the Hungarian cockade :D
I'm not sure I've ever heard anyone refer to a prism as a "square"? Is this a thing people say?
Haven't heard it either. People do call 3 dimensional objects squares, like a square of cardboard, but that seems to be just because they approximate 2 dimensional objects.
Are there other theories that have been proven equivalent to measure theory for probabilities?
You have been responded!
That's part of the answer: en.wikipedia.org/wiki/Free_probability
I had no idea that probability has had so many rigors apply to it so relatively recently.
"saying something has probability 0, does not mean it's impossible"... yes it does; that is exactly what it means.
I think pseudo-random number generators are interesting too. Have people studied the probability of those generators actually producing "random" numbers?
Kelsey, ,great episode, and thanks for pointing out the man-bun should be no more. Unless you're Toshiro Mifune!
suppose you are playing a game in which you win $15 if heads come or lose $10 if tails come. if you toss the coin 10 times then you might not win anything or lose the money but as you toss the coin many times (say 100) then the result will get closer the probabilistic value i.e 50/100=1/2 and you will win money
so in a bet which gives you advantage it is advisable to play maximum number of times possible
"Infinitesimally small" doesn't mean "exactly a point". A point doesn't have an size, whereas an infinitesimal area does.
I wouldn't honestly place Pascal's Wager in the probability discussion, as he was attempting more to use rational thinking about probability and applying it to theological thought with a particular bias in the first place.
Pascal's wager is logically flawed. It only works if there is only one possible god that can be imagined. But in fact quite the opposite is the case, there is an infinity of possible imaginable gods. So any probabilistic reasoning has to take all of them into account and many are mutually exclusive - what earns you reward with one may earn you punishment with another.
Pascal probably wasn't aware of the prisoners' dilemma in game theory when he thought about god.
Best part about probability is that it isn't statistics
Remember guys: If it ain't statistics, don't fix it
Linkmario Fan a lot of stats is in probability though.
A paradox arises from requesting an infinitely small area from a granular universe.
I'm still not convinced of the solution (or the logic) of that famous problem:
"You choose between 3 doors, one has a price. After you choose someone opens a door without a price among the ones you didnt choose. You then have the choice of changing your door to the remaining door. Is it wise to change it?"
So, the answer is "yes" because of the probability of the first choice "1/3 chance of getting the winner door" and "2/3 chance of the winner door being among the others". However, my question would be: Since theres 100% chance that there is a empty door among the ones no choosen, and the revealed door is always an empty one (it is not a random choice), why is the probability of the winner door added to the non-choosed door instead of removing it from the spectrum? In the end, I have a door and theres another door, one is a winner. shouldnt it be a 1/2 chance?
+Francisco Roy De Mare You can't ignore the history of the scenario. If I ask you what the probability is that the top card in a 52 card standard deck is the 3 of clubs, the answer is obviously 1/52. But if I show you the bottom card in that deck and it is the 3 of clubs, then ask you what the probability is that the top card is the 3 of clubs, the probability is clearly 0. In both cases you have a 52 card deck, but the fact that the current state is the same in both situations is irrelevant.
You have 2 doors, but those 2 doors took different paths to get where they are. The door you chose has a 1/3 chance of having the prize behind it. It had a 1/3 chance when you chose it and, as far as you know, nothing has happened to it or the prize since then. The 2 doors you didn't choose had a 2/3 chance of having the prize behind one of them. That too has not changed. The 2 doors you didn't choose still have a 2/3 chance of having the prize behind one of them. But, because you know for certain that the prize is not behind the door that was opened, the entire 2/3 chance of having the prize is with the other door.
Yes, I understand that. And probably a sample test (repeat this scenario 10000 times would confirm), but I still cant see the objective logical demostration.
Your card example makes no sense, its like hes showing me that one the other doors has the price.
Like I said, the revelead door is ALWAYS empty, its not random. So theres only 2 doors in the game, one with price and one without.
Its very counter intuitive, thats why the problem is famous (?)
My problem is that if I was watching the show the previous day, my first choice of door is irrelevant, whatever I choose the winning door or a empty door, Im still picking between two at the end. So "Is it wise to change it?" makes no sense for me, 2 doors, 1 choice... maybe "Is it more likely you get the right door twice?" 3 doors, 2 choices.
Pascal's Wager assumes he believes in the correct God. But there are so many!
I'm interested in the use of elementary probability to aid in rational decision making.
As a physicist, I'm all for Solovay Model. Down with axiom of election! Just joking, of course.
The Results get more accurate as we increase the number of experiments.
be there or be square came about because if you're not there you're not "a round". also it rhymes
I completely missed this when it first appeared due to what I think was insanity. I'm not sure the insanity is over but I just wanted to say that I think there is a hidden secret in coin flipping. It might even be tied to universe being a simulation. But remember, I'm not sure the insanity is over.
If i'm waiting for somone to arrive for an appointment how long do i wait after the agreed time before i cut my losses?
A next-door-neighbour subject is that of reasoning given only probabilities and events. That is, given that event A has happened, and a bunch of probabilities, what is the probability of event B?
Please more videos like this!
You forgot "public gathering place" for "square"
OW! my brain....ok take a break come back OW! my friggen brain.
This channel always motivates me to study more math, sometimes school cant keep me interested enough, although I love math.
Kolmogorov is easily one of the top 3 mathematicians of all time.
My favorite part of probability?
AdDiNG FrACtiOnS
If probability is rooted in measure theory, how does bayes theorem relate to measure theory?
Bayes theorem arises naturally (ot, to be more precise, it is a natural definition for the intuitive notion of a conditional probability).
Assume you have a probability measure space X (meaning you defined a measure/"size" P for subsets of X and X has measure/probability 1, ergo P(x)=1) and a subset A of X. Then you natually also get a a new measure for subsets B of X by taking P(intersection of A and B).
This new measure has one problem though: The new measure of X is no longer 1, but the smaller P(intersection of A and X)=P(A). To once again get a probability of 1 for X, you need to "normalize" the measure by the factor P(A).
So in total the new measure then reads "new measure of B" is "P(intersection of A and B)/P(A)".
This new measure is called the probability of B under the condition A.
As a stochastic explanation: A and B are the events (as subsets of the set of all results). You know know that A already happened, so you only need to consider points in A and want to know the measure of points in B. So you take set of all the points who lie in both A and B the intersection). But because you want to model the fact that A already happened, A must have probability 1, so you need to normalize the whole thing
Raphael Schmidpeter
Ah! Not only a prompt response but a clear one. Cheers!
Could it not have been based on measure theory? Would a different initial aproach make certain answers easier to reach? Would it make it apply more directly to quantum computing?
Did I just notice a mistake on the first time se showed the concentric "cuarter" when it was a half?
4:43 What about squares that are places in a city? Like a town's square or times square?
When you start talking about the dart board and hitting a single point on that dart board: you need to properly define that point. If you expressly mean hitting exactly a 0-dimensional point and no other points around it, then that is impossible...but it's also somewhat redundant to mention.
If however, you mean that each point is infinitesimal in size (and this is typically what people do mean, as they are dividing the board into an infinite number of sections), and that the dart head is of a comparable size (i.e. also infinitesimal as already stated in the video), then the probability of hitting any specific point becomes the proportion of that point to the overall size of the board which is infinitesimal and clearly non-zero. Applying a real number field to this type of maths where you either take limits or involve certain types of infinities/infinitesimals is erroneous. Instead you should be applying a number field that includes these things as numbers (e.g. the surreal number field), which would lead to much less confusion over such things as inaccurate approximations are no longer used.
Practically when you hit anywhere on dart board, you are simultaneously hitting a Planck area of points. So the probability of hitting any given point must be 1/(No. of plank Areas on the dart board) which is non zero. Just saying...
How are you defining Planck area? You seem to treating Planck area as an object, rather than a unit of measure. If Planck area is a unit of measure, each dart lands in uncountably many circular disks each with a area of 1 Planck area.
Steve's Mathy Stuff, OK let me rephrase it. Probability of hitting any given point must be (Area of 1 Plank Area)/(Area of the dart board).
Why? If the probability that the dart will hit point A is (Area of 1 Plank Area)/(Area of the dart board), then hitting A must mean hitting the unique subset of the dartboard with area 1 Planck that contains A But there is no such subset. Uncountably many subsets of the dartboard have area 1 Planck and contain A.
You could select a finite subset of all the mathematical points on the board and designate the disks centered on those points as physical points. (Mathematical points in overlapping disks can be assigned to one or the other somehow.) If you chose wisely, the finite set of k disks covers the board and the probability of hitting a point could be said to be 1/k. k would always be larger than (Area of the dart board)/(Area of 1 Plank Area) due to overlap.
There are uncountably many finite subsets of the set of points on the dartboard. For any of those, there is always one that makes the probability distribution more uniform. (That's based on my method of assigning mathematical points to physical ones in ambiguous cases. There might be an assignment method that does not skew the distribution.) Do we have any reason to believe that any one set is the right one?
My logic is when the dart is actually hitting any point within the Planck area circle around point A, it is also simultaneously hitting point A. Any measurement system will give same response for the events of dart hitting Point A & dart hitting some other point within the Planck area circle around Point A. So dart hitting any Point within the Planck area circle around point A should also be considered as hitting Point A.
Let r be the radius of a circle with an area equal to 1 Planck area. Suppose point B is hit by the dart. The dart is treated as if it hit point A, a point within r of B. But uncountably many points are within r of B. Which one is point A?
If Pascal believed that having a false belief is no great loss I can no longer call him a scientist or mathematician.
+You Tube Two points. First, many scientists and mathematicians held and hold absurd beliefs. Second, I think Pascal only said the loss was small in comparison to the possible gain. I think if Pascal had understood measure theory he might have reached a different conclusion. He assumed without proof that, in the absence of an afterlife, his lifespan had finite measure.
Well he's either laughing at you or doesn't care, you on the other hand look stupid in both cases. Scientists are just people, although science pretends to be an absolute truth it'll always be just a concept
Bayes theorem
Are there other axiomatizations of Probability that involve something other than measure theoretic constructs?
Very interesting the kind how to treat probabillity as measure. I would like to launch a problem in the channel about measure. If I divid a pizza in equal pieces, the number dosen't matter but to make it easy, in 8 equal parts, wich part is the biggest? I call it the paradox of the stretch' time.
Wow, now I can understand more Monte Carlo.
Isn't the probability of hitting any point on the dart board actually 1/infinity ? Seems like asserting that the probability is 0 is the same as asserting that 1/infinity = 0
that's because any finite number divided by any infinity is equal to zero. At first glance it might seem paradoxical until you realize we are talking about a freakin' infinity....
But you ought to remember that what you're calling "infinity" and "zero" in this case have a product equal to 1.
+Graham Henry, I agree. As I understand it, limit functions are needed to define "infinity" , and to define or evaluate "1/infinity". But this seems to get lost to the confusing vagaries of so called "Real Numbers". It's perhaps more clear to stay within a 2-Dimensional space based on areas, and then apply limits if necessary.
Physically there's a finite number of points with each atom on the surface of the board representing a point. So there's 1/# of atoms.
And if you want to be practical and not as precise, just split the board into hundreds of equal parts and throw the dart. Now the probability is 1/# of equal parts, which is more useful
I don't understand 7:15 , the transition from 1/2 1/2 = to zero because the area is zero?... I don't understand
I can't get over the idea that the probability of hit a point is Zero!!
probability of a probability is that its a probability
I've always thought probabillity isn't real. If we look at the darts example, we can (in theory) accurately calculate where a dart will hit if we take into consideration the relevent data (Player's strength, dart's direction and weight distribution, wind, coriolis effect, etc).
I do however find probability theory convenient for engineering.
Quantum mechanics, which determines the behavior of the tiny particles that make up everything, is fundamentally probabilistic. Wave functions and particle interactions in quantum mechanics and quantum field theory are most accurately described as a set of probabilities, and are capable of describing nature more accurately than classical physics. And although the randomness diminishes as the scale gets bigger, it's still there.
In other words, there is a 99.99999999999999999999999999999999999999999999999999999999999999999999999999999% (perhaps even higher) chance that the dart will land in the spot predicted by factoring in every aspect of the air around it, the way it was thrown, the mass distribution of the dart, and the gravity of every last particle around the dart, but you cannot say with complete absolute certainty that it will land in the spot you think it will. Also, Heisenberg's uncertainty principle says that you cannot know the position or the momentum of a particle with exact certainty, and the more accurately you know its momentum the less accurately you can know its position. These effects diminish at larger scales because the relation between these uncertainties only becomes relevant at the atomic scale.
So probability isn't real in an absolute sense when dealing with classical physics, which is mostly relevant on the scale of darts and scoreboards, but in the complete picture of the fundamental workings of the universe, probability is "real" in the sense that you are talking about as there are things that can only be described (or are most accurately described) as probabilities.
Did anyone else see this an hour ago but couldn't watch it?
yes that happened to me
understanding nntaleb's tweets and MOOCs
the area of the dart is assumed infinitally small ..not zero...so probability of hitting the board at a point is not zero at all....whats the paradoxical about that.?
No the radius of the point is zero not just infinitely small. Remember you're hitting at an exact point so the radius has to be zero