# conditional probabilities in chess

Let's say I am analyzing certain chess positions of a chess player (Magnus Carlsen, Lichess games when playing as white).

We know that the 'unconditional probability' (empirically speaking) of this chess player to win a single game is around 72%, 3511 won games out of 4873.

Let's say we observe that when he plays against a Scandinavian game (1.e4 d5, 2.exd5 Qxd5) he has "higher probabilities" of winning (80.3%, 57 won games out of 71)

In the same regard, we know that when he plays against an Alekhine (1.e4 Nf6, 2.e5 Nd5) he has "lower probabilities" of winning (65.3%, 47 won games out of 72)

Let's call these last 2 examples the 'conditional probabilities'.

The question here is, how can we know that there is indeed a higher/lower probability of winning conditional in some positions and not that it was an output by random chance? I thought on testing the 'conditional probabilities' against the 'unconditional probability' of winning a game (ie. with a bayesian approach, where the beta is located in the unconditional, and we simulate the outputs of 72 games). But isn't that a problem because my unconditional probability also depends on the output of those 72 games?

Like, isn't fixing the 'unconditional probability' a wrong approach because it is also a random variable? Even though we have much more samples than in the other positions.

• One standard solution is to regress the wins (using logistic regression) against the type of opening, using planned post hoc comparisons among the openings (or, as an omnibus method, applying something like Tukey's HSD to the results). Simulation is neither needed nor helpful. I don't follow your final remarks, because they appear to reverse causality and logic: the outcomes of the games depend on the (hypothetical) probabilities, not the other way around!
– whuber
Commented May 3 at 16:15

I have concerns about this, but a simple approach is to regress the win/loss outcomes against the types of games, such as with a logistic regression. This then turns the problem into what might be a more familiar, ANOVA-type of situation, where you can assess which types of games result in higher or lower probabilities of victory.

A concern I have is that time plays into this. For instance, the player probably got better and better at chess in general as he continued to play. If a type of game only debuted midway through the player's career, then it is only natural to expect the player to win more against that type of game. Fortunately, however, there are ways of icorporating time into regressions, and you may consider doing so.

• I thought on the same, but I think in general, because of the nature of the pairing system (using elo) those differences should disappear. Let's say, if the player was 'bad' at the beginning and he was only playing a french defence, his probabilities of winning in any position were still independent of his elo, because the people he is playing against are also expected to be bad at those positions (they could not exploit it as easily as higher level players.) Commented May 6 at 7:49