IS CHESS A GAME OF CHANCE? Classical vs Frequentist vs Bayesian Probability

what is probability exactly what is chance what is luck if you shuffle cards or roll dice or flip coins then we all agreed there's a probabilistic element to it but what about a game like chess a game of perfect information where everyone can see anything and there's no explicit probabilistic dependence you might think obviously there's no chance involved but consider this if i play people of about my skill level then the amount of time i win the amount of time i lose is about the same it's kind of 50 50 ignoring draws that sounds kind of probabilistic in this video i want to talk about the different types of probability different perspectives on probability and how probability can enter our lies a little bit more often than we might think my thanks once again to brilliant for sponsoring today's video more about them at the end of the video let's begin with something called classical or theoretical probability this is a normal deck of card with an equal number of 13 hearts 13 clubs 13 diamonds and 13 spades because they're all equally likely if i draw one off the top at random then there's a one quarter chance that it's going to be a heart in classical probability we write this as the probability of drawing a heart is just the total number of hearts available which is 13 divided by the total number of cards which is 52 and that's equal to one quarter more generally classical probability applies in a scenario where you have a finite number of options that are all equally likely like drawing an individual card is all equally likely and then the probability of a particular event like drawing a heart here we just take a ratio of the total number of times that that particular event could occur in our case join a heart and then divide it out by the total number of possible events in this case the total number of cards so classical probability is great in fact i have a whole sequence of videos on classical probability in my discrete playlist you can check it out down in the description but it's limited to scenarios where you understand exactly what all the options are and what the number of all the options are and that those options are finite so what if that isn't true take the same deck before but i've done some new things i've chosen some cards on the top that i want to get rid of and i'm actually going to put them away and now we have this new deck it no longer has 52 cards but here's the thing you don't know what cards are in it you can no longer use classical probability you don't know how many hearts and clubs and diamonds and spades there are well one thing we could do is we could keep on doing our approach of shuffling and let's see what we get here on the top we get well a heart and well we could keep on going this way we could do another shuffle we could do another cutting and we could see that what we have this time it's a diamond so imagine you do this like a thousand times and perhaps you collect this data 342 hearts 337 clubs 321 diamonds and zero spates those numbers look an awful lot like i just removed all the spades and it's just an equal number of clubs hearts and diamonds but probabilistically when you're drawing them they won't be exactly 333 every time it'll be a little bit distributed this is the frequentest approach to probability it says i'm going to collect a whole bunch of different empirical data points i'll be randomly selecting cards over and over and over again if i didn't like this if i wasn't convinced that it was 1 3 1 3 1 3 and 0 i could go to 10 000 draws and it would probably be a lot closer to one third one third one third zero if that's what the deck of cards actually is for any number of experiments that you do you could still be making a mistake i mean there might be a spade in this deck it's just that you've been very unlucky over a thousand draws to have never gotten it probably that's not the case but it's possible so the frequentest approach basically is defining probability as a limit it's imagining you were taking these experiments over and over and over and over again and in the limit as n gets large the ratio of what you get the number of events that you're trying to measure divided by the total number of trials that you've taken the total number of experiments well that is going to become the frequentest probability so in classical probability you sort of know everything about your scenario you you know everything about the deck of cards and how many cards it has and you're sort of doing this a priority calculation of what the probability should be but in the frequentest approach this is great for when you have an imperfect scenario and you don't know what's going on then you collect a whole bunch of data and you say well okay now i can figure out what the probability is going to be based on my trials if you have something like a six-sided dice where you do know it both of these approaches the classical and the frequent distal line the frequentist as you do a large number of trials is gonna see that each side comes up about one-sixth of the time and in the classical probability perspective where you know that there's a six sided dice that are equally likely again you'll say each side will come up one sixth of the time but the frequentist approach is itself limited for example what if you can't just do an enormous number of trials like if you're going to try to predict whether your tesla stock is going to go up tomorrow you can't just take a large number of trials of many tomorrows you've only got one and you still have to make a prediction the third approach to probability is going to be called the bayesian approach to probability i've cut the deck one more time and now i'm going to draw the top card except this time i'm not going to show you i can see what it is but you can't and so now you can ask the question what is the probability that this card is a heart the point here is that it's subjective from my subjective perspective where i can see what it is i have more information than you do and i know what card it is but you have a different amount of information you might have a prior belief that there's a one-quarter probability this is a heart but then if i flip it over and reveal that it is indeed a heart you update your probabilities and now you have what is called a posterior probability and you say there's a 100 chance of it being a heart if i do it again and reveal a new card well the frequentest would take a different perspective than the bayesians because the frequencies wouldn't say that there's a one-quarter probability they'd say well hold on the card is chosen the card is fixed there's nothing probabilistic about it it's 100 hearts or zero percent hearts the frequentists have a different philosophical viewpoint here they might think that their attempt to explain the world by suggesting that it was a heart is something that in a frequentist approach if you did this many trials over would be a correct prediction about this objectively right answer one quarter of the time in contrast the patient is totally happy thinking that things depend on the amount of information that you have that it's subjective and it's not until you gain the new information that this is a diamond that you update your probability you say there's a zero percent chance that it's a heart we just really want to think of the bayesian perspective as you're having these prior beliefs and you're going to update them to get your posterior beliefs as you gain more information about the world okay so let's return to chess a game where there is no shuffling of cards there is no rolling of dice a game where both players can look at the board and have perfect information about all the data on the board so how come then that against equally rated players your chance of winning your chance of losing are about the same if you're a frequentist and you'd see this sort of 50 50 behavior you'd say well okay that does mean you have a 50 probability of winning any particular game if you're in a more complicated position where say your entire piece up which at the level i play at which is particularly great i like having fun with chess but i'm not particularly good but nevertheless if you're up a whole piece it's very very likely that you're going to win and so a frequentist can say well of the games of which i have been up a piece i win with a probability of say 95 from a vision perspective you might go into the game thinking that about half the time that you're going to win and then as you gain new information you're always updating your probabilistic viewpoint as to whether or not you're going to win and it doesn't even have to be that like a new piece is moved and that's what updates your impression it could just be that you're sitting here and you're calculating away and then you realize what the winning idea is you do have a force checkmate and then you update your probability that you're going to win accordingly but where exactly does this probabilistic way of thinking about a clearly game of skill like chess even come from well the idea is about bounded rationality that is chess has about 10 to the 120 different possible games it's unfathomably large for a human or even for a computer to be able to just mass calculate everything we have a sort of bounded rationality which is we can be rational we can do calculations a little bit into the future but then we lose our ability to do that a common model for how people play chess is you might start at some particular situation and then you've got really three different options that you're considering maybe there's many other options but they're all just sort of either obviously bad or don't really do anything you just don't bother focusing on them and then if you have these three options perhaps your opponent has three options they consider in your response to your three options and now there's sort of nine possible little trees and then you might have three things that you're going to consider each of the nine things that they consider you can see how even with just a very small amount of back and forth the amount of calculations you have to do start becoming really large the way this really works for probably both humans and sometimes explicitly for computer engines is that you take these trees and you prune them down and there's a whole bunch of options you just like immediately don't even think about because they're just sort of obviously bad or obviously pointless and you cut out many of these as well and you're really maybe only considering a choice between just a couple different moves a couple different paths and trying to evaluate whether you think those are good imagine for simplicity's sake we can both only see three moves into the future so we both think we've done a good job we've looked as far as we're both able to look with our bounded rationality and then it turns out that one move further than we can think there's a huge advantage for one player or the other it just results in the situation like say a fork where you can easily take one of two pieces and gives a massive advantage for one of the players this is the type of scenario where there's something a bit probabilistic to chess at the limits of your bounded rationality it can result in situations that sometimes are fortuitous to you and sometimes ones that really cause a lot of problems now i want to be completely clear chess overwhelmingly is a game of skill somebody who's even a few hundred rating points better than i am is just gonna reliably completely decimate me over and over and over and over again and so would i to someone who was a few hundred rating points beneath me but when you're matched with someone randomly of your own skill level and the skill becomes equalized then some of these sort of probabilistic elements like what happens beyond your region of bounded rationality starts to dominate who wins a specific single game humans i think tend to underestimate the role that probability plays in their lives particularly if things are going well you probably want to attribute it to your hard work and your skill and your dedication maybe if things are going really poorly you say okay well then i just had bad luck but i think the truth is that probabilistic elements actually influence our lives on balance a lot more than people tend to think about and i think that moving beyond just sort of a classical view of what probability is to a view where it's either a frequentest approach and you're imagining many things happening over many trials or a bayesian approach where you're updating your probabilities as new information comes in that this is going to give a broader world view about how probability affects our lives i think sometimes math can feel a little bit like magic if you draw a random card you can shuffle it back in and then i can find it again easily but magic tricks aren't magic they can be understood and the same is true of math we can demystify math and that is why i am so proud to be sponsored once again by brilliant we've talked a lot about probability in this video and they have a really cool sequence of courses on probability that take you from the basic fundamentals all to really cool applications like the blockchain i actually had a ton of fun with their perplexing probability course which is all these cool lessons like the classic perplexing probability example of the monty hall problem where there are three doors only one of which has a prize behind it and you have to try and win it whether you're a magician or a math magician to really master your craft requires more than just knowing how the trick works you have to actually practice it you have to get your hands dirty and in my opinion this is where brilliant really shines their courses are incredibly interactive you're not just taught the material you get to practice it you get to play around you get to have feedback on whether you're understanding the content as a professor i know this is just excellent pedagogy and why i'm so proud to be sponsored by brilliant so go to brilliant.org trevor bassett sign up for free and the first 200 people to use that link are going to get 20 off an annual premium subscription and with that if you have any questions about this video leave them down in the comments below and we'll do some more math in the next video

Channel: Dr. Trefor Bazett

Views: 99,594

Keywords: Solution, Example, math, intro to probability, is chess a game of skill, is hess a game of chance, bayesian analysis, bayesian inference, bayesian probability, bayes' theorem, frequentist probability, subjective probability, emperical probability, deck of cards, classical probability, theoretical probability, a priori probability, chess

Id: dejy4PPCFHY

Length: 13min 26sec (806 seconds)

Published: Wed Nov 24 2021