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

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What, exactly, is probability? In this video we will see a few different perspectives on chance, the classical or a priori viewpoint, the frequentist or empirical viewpoint, and finally the Bayesian or subjective viewpoint. We’re even going to consider the game of chess, what appears to be overwhelmingly a game of skill, but which the limits of our bounded rationality still results in probabilistic elements.

0:00 Intro to Probability
0:50 Classical Probability
2:09 Frequentist Probability
5:18 Bayesian Probability
7:14 Is Chess a game of chance?
11:05 Underestimate the role of chance

Classical Probability: This is when we have a finite set of equally likely possibilities, and thus the probability of an event A is just the number of times A occurs out of all the possibilities. This is great when we know everything about a situation like a normal deck of cards.

Frequentist Probability: This is when we do emperical studies and see how frequently event A occurs, particularly in the limit as we do a large number of studies. This is great when we don’t know a priori exactly what is going on like some deck with an unknown number of cards removed, but the downside is we need to be able to do a large number of trials.

Bayesian Probability: This is when we begin with a prior probability and when we gain new information we update our worldview (often using Bayes’ Theorem) to get a posterior probability. This viewpoint is subjective because you and I may get different results given different information and different even if we had the same prior probability.


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  1. Great explanations of the three statistical perspectives! Thank you for this fresh and hands-on video.
    Do you play 1.e4 or 1.d4?

  2. I love everything about this video!

  3. There are some wrong or misleading points made in this video though the question is interesting. First, chess is a calculation. In principle there's an outcome with perfect play. Therefore theoretically you can't improve your position against perfect play. The number 10 to the 120th is Shannon's estimate. However if you get rid of ridiculous moves it's closer to 10 to the 40th. Second, the win rate is not equal for white and black. Unless chess is a draw when completely solved. It's thought having the first move gives white an edge and that's largely true over chess history. But the draw rate is climbing, probably due to engines and databases. Third, 200 rating points predicts 75% ‐ 25% score for the higher rated player. I would not say that means the lower rated player has no chance. But unless you've played in tournaments against serious players it might seem that way. I've played friends who said they were good who didn't know the en passant rule. When they argued, my Bayesian Probability went to 100%.
    I definitely think there is a probabilistic element to real world chess. In fact, players are often at the mercy of the pairings – which color, when, the opening – even against the same player. More interesting is choosing a move. Magnus Carlsen will often chose a slightly worse move if it causes his opponent to have more choices, some of which are bad, and waste time. Of course Magnus usually blunder checks those choices. Having a chess engine that used this probabilistic method would be a great learning tool. Not that it would change the engine's win rate against humans.

  4. Nonsense. Chess only appears to be a game of chance because we lack the computing power to resolve it, and must resort to approximations. One day, perhaps with quantum computing, we will determine all chess outcomes as completely as Tic-tac-toe, and will know exactly the sequence to win or draw every time.

    Chess will continue to be entertaining for amateurs, but will have lost interest to masters.

  5. Bayesian probability is fine, but when you use it to argue that reality itself is subjective my mind gets f*cked. Talking about Quantum Bayesenism here.

  6. I think there are 2 decision points in playing chess that are more or less a guess. That might be based on a gut feeling or experience (human) or a mathematical approach of scoring the state of the chess board for that moment (computer). Those 2 decisions are made on what to prune and on how to rate the expected chess board after the expected outcome of a couple of moves. For a computer a Bayesian approach for rating the states might be out of reach because there are simply to many variables to take into account. I strongly believe Bayesian would be the superior choice if possible. However a frequentist approach would be more likely feasible. You could research what kind of statistics are important to keep track off and let a computer play millions of virtual chess games to fill in the stats. That way it might be possible to rate a situation based on a frequentist approach. That frequentists approach should be considered 'non-stationary' and trained continuously. Otherwise it would eventually start to loose games if the opponent figures out the general strategy.

  7. If you flip a coin 15 times you will get a wide variance in the number of heads. Yet somehow the media forces basketball players to say oh this or that was happening in the game to explain the outcome (which often there is some truth to) but sometimes it's really just simple luck – the ball just didn't go in.

  8. I'm curious what is your ELO is? And if this thinking about probability could improve your chess skill? For example, when I assume that my opponent calculates equal moves ahead as I do, it motivates me to think longer/more focused.

  9. My initial reaction to this video is that it's a bait-and-switch. "Is chess a game of chance?" No…It's absolutely a game of perfect information, and I don't see why it matters that the state-space is unfathomably large for a human to grasp.

    The reason I say it's a bait-and-switch is that I think the Dr. is actually asking a different question: "When humans play chess, are outcomes probabilistic?". To this, I 100% agree; that's why we still play it. It's a skill based game, but given two similarly rated opponents, anyone can win. That's why chess tournaments aren't settled with a best-of-one format.

    Philosophically, we can toy with the idea that game of chess is probabilistic because the state space is, at least for now, incalculably large, and therefore we humans that play it have no choice but to imitate a random walk through that state space. However, I don't see it as a particularly fruitful line of thought.

  10. Very helpful video. Just first learned about the Bayesian network and the belief network in class without learning Bayesian probability before. This video clarifies the difference between classical and Bayesian probability for me.

  11. But isn't this how the rating system is defined? If two players have equal ratings, then they are calculated to have equal chances of winning against each other. All you are saying by noting that two equally rated players have equal chances of winning is that the ratings system work well, and the rating system works well because chess is not a stochastic game. In order to score players we need a sample of their performances against other rated players. Over a large enough sample, the rating given to a player will converge on a value that has a certain predictive power. The sample size needed to achieve this power will simply be smaller for chess than a game like backgammon, where dice are involved.

    If we take a game of pure chance – like tossing a coin – you could sample indefinitely and still have no information on who is the "better" player.

    There are reasons why one chess player has a higher rating than another, one of which is the experience to know how games will pan out outside their (and their opponents) move trees. Knowing that a particular position will trade down into a favourable end game, for example. So it is not really true to say that chess becomes stochastic at a certain tree depth. This is one reason that the best human players held out against engines for so long.

  12. Your Win/Lose rate might be almost 50/50 but that is not true with my winrate (13% more wins tha losses) and there are way more extreme examples. People with 2 to 1 win to loss rates etc.

  13. If you don't like the Bayesian saying the probability of something depends on who you are and what you know, would it also be correct to frame it as "in situations like this, where I know what I know now, I expect this guess to be correct X% of the time"?

  14. The explanation completely disregards many of the most important aspects of the game of chess, especially opening theory and endgames study (both of which considerably reduce what you call "bounded rationality"). That's why a player like Magnus Carlsen can simplify almost any opening into a known endgame and calculate the winning moves (check all the available moves to find the best ones at a depth much, much higher than 3). In a classical (ie long) game between strong players, there's almost no "bounded rationality". Novelties (ie moves that aren't part of opening theory) trigger calculations that will almost always lead to a known ending pattern. Chess only is a game of chance in quicker time formats or between amateurs.

  15. Check out the work of Ole Peters, you might find it interesting

  16. Slightly off-topic, but you'd expect to have a roughly 50/50 win/lose rate against people of your skill level because that effectively is the definition of "people of your skill level".

  17. Classical probability should always be modified by the increased frequency of what current and past outcomes have resulted because you are never given complete assurances that the data points and elements within a game environment have stayed the same. The trend is your friend until you sense something is different or changed then your reliance on past outcomes no longer applies.

  18. There's certainly some probability, but making a chess move is a mixture of intuition and calculation. Even if both players can only calculate 3 moves ahead, at every level the intuitive element of discerning which move leads to a more sound position can dominate at the highest levels of chess as well as lower levels. So it's hard to say that it was chance that the move I selected ended up leading to an advantage that I couldn't calculate all the way to, because even though I couldn't calculate my way there the move itself wasn't random but based on some pattern recognition about what makes a better position on the board.

  19. Basically this was an add for what you are hawking. You made no relevant conclusion about chess….. Thumbs down…😁

  20. That was a great introduction to some interesting math that didn’t go too deep into the details. Just enough to make you interested enough to want to learn more.

  21. The myth of meritocracy is more like a legend– effort and ability play a major role, but somewhere down the line we forgot that randomness/chance/probability play a lesser, yet ever present role in outcome. I believe societies benefit from pushing the "bound of rationality" for individuals, allowing those willing to see further into their potential future.

  22. I believe there are five possibilities here that are inherent to the game of chess per se (considering an ideal situation where both players never fail and the rules of chess remain in its current standard). 1st it's the possibility that who starts (the white) has the victorious advantage, so the whites always win. The 2nd possibility is that the player who responds to the 1st move has a victorious advantage, so the blacks always win. The 3rd and 4th possibilities just add more depth to.the first two possibilities. The 3rd possibility is that who begins first is not influential to the victory, however as the distribution of the armies in the board is not perfectly simetrical, working as the reverse/mirror image of the other, the army whose pieces occupy that sector of the board where the respective king occupies the black house (the whites) always wins, regardless if the whites begin 1st or not. The 4th possibility is that the black army always wins because their pieces occupy houses with reverse colors to the white (the black king is in the white house), and that's the victorious advantage. And finally the order of who makes the 1st move, and the initial positions of the armies on the board have no influence in the outcome, and both armies are equally powerful, meaning that the chess game is no more than a hyper tic tac toe game, faded to stalemate unless one of the players fail. As the chess game is so complex we don't know if thr initial position of the pieces, or who makes the 1st move is influential or not to the victory, so we can't know (till now) if the chess game is the hyper tic tac toe game or not. Which leads us to another question, would be all known skill games hyper tic tac toe games whose winners are just harvesting the opponent's errors? That means in a skill game between same skill level opponents the real winner is the biorithm of the winner.

  23. Its not really chance. In theory, white has an overwhelming advantage, and the best that black should hope for is a draw. But that only exists in a vacuum where both players are of equal skill. Kind of like playing against your exact clone.

  24. This was a great video 🙂 I don't usually like edu-tainment but this one I think had sufficient depth without being overwhelming. I your point about bounded rationality, very well-spoken and intuitive.

  25. You lost the track on explaining Bayesian probability around 7:00 when you delved off into chess without continuing to relate Bayesian's application to it.

  26. I guarantee you that chess is not a game of chance. It’s very low variance, there is no hidden information, and a person with proper technique will beat a weaker player, far far more often than in something like poker.

    I’m an expert level player, about 1800 ELO and I guarantee Magnus Carlsen beats me 1000 times out of 1000. likewise I will beat a player who knows just the moves 1000 times out of 1000. There is simply a massive difference in technique between a beginner, intermediate player, and a grandmaster.

  27. What is the probability that the googol'th decimal digit of π is even?

  28. but the board doesn't contain all the information in the system?

  29. I think I'd begin a discussion of this issue with describing the game rock-paper-scissors, to explain how chance could enter a game like Chess.

  30. When aliens arrive, what is the probability they will greet us with “All your Bayes are belong to us?”

  31. I don't see how classical probability fails to analyze the situation where we've removed N cards from the deck.

    Can't we just construct the set of all decks with N cards removed, and calculate the number of possible scenarios where we flip one card from the 52-N cards, and then count the number of situations where that card is a heart, and divide as usual?

  32. Human chess at any level necessarily has luck. You, in fact, don't have complete information. It's not the game rules and placement of pieces that wholly define the variables. Tons of variables remain that pertain to yours and your opponent's current psychological state. Everything from being in a (win/loss) streak, to pressing personal issues to performance enhancement drugs (even caffeine: properly dosed, overdosed, underdosed). Engine/AI chess is a different kind of game.

  33. I think what is called gut feeling in chess is derived off of probabilistic way in which subconscious mind is thinking. The ideas in game after many hours of practice have an abstract feel to them.

  34. Fun fact: The heuristic, being up a piece in chess is a good winning indication, held up for a long time. Then AlphaZero came along and proofed that assumption wrong. It won many games because of its ability to sacrifice piece in order to get ahead. Great Video!

  35. just went 1-3 in a chess tournament, losing to lower rateds…elo onward!!!

  36. There’s a chance that someone plays a REALLY bad move or misses a REALLY good move. The distribution, over all your moves, of your chosen move relative to the best move, is a normal distribution. An average player will most often choose a move that isn’t the best, but is not detrimental. The distribution skews toward the lower or latter ends dependent on skill level.

    That’s speculation but I’m pretty sure it’s true and I’d bet it’s a normal distribution because of the central limit theorem.

  37. This title makes no Sense and you know it, Chess is not a game of Chance because even if you wax philosophical like you did everyone knows what "A game of chance" means, this isn't clever it's just dumb.

  38. There's also noise and bias attributable to each person. For example, daily differences in whether you are mentally more or less efficient due to what you ate for breakfast, whether you slept, etc., can make a big difference in how well you practice your skill. Same goes for whether you are hungry, or in a hurry.

  39. What distinguishes a better player from a worse one is of course their greater insight and intelligence, but that's not the whole story. Relentlessness and consistency are among the greatest predictors of success. That means even if you're technically more proficient than someone else, they can still outcompete you in the long run if they push themselves significantly harder.
    In a chess competition that's like turning a losing game into a win by continuously building threats, while the opponent ignores the problem until it's too late.

  40. of coursethis is the case with every activity in which humans engage in ever. if I'm a three point shooter and am in a first to 10 3 point contest with another 3 point shooter with the exact same FG % as me, whoever wins is chance. its 50-50. that's the very definition of "opponent of equal skill" otherwise it'd be 60-40.

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