Estimating Win Probability from Score in Best-Of Series Consider a best-of-31 game where team A won 16 times and team B won 14 times. Assuming a rematch in which all conditions are the same, is there any way to infer (or guesstimate) what the probability of team A winning is?
My attempt:
I used team A's score and the total number of rounds to find the probability of A winning a single given round, i.e. A score / (A score + B score). In this case it was 0.533. Then, I wrote a Python script which gives the probability of a team winning a best-of series, given their probability of winning a given round. The results for this example using 0.533 was 63.2%.
This seems rather high for a close match, and indeed using a score of 16:12, team A has a massive probability of winning of 80%.
Is inferring a probability from the score complete nonsense, and is there a better way? Can my attempt be considered useful?
 A: $16/30$ is the maximum likelihood estimate for the single game win probability of team A given your data, so it is very reasonable to use this value. In fact, without additional data or assumptions, it is the only reasonable estimate.
Team A's probability of winning the match is equal to the probability that the binomial random variable $X \sim B(n, p)$ is greater than or equal to $16$, for $n=31$ and $p=16/30$:
$$
\Pr(X \geq 16) = \sum_{k=16}^{31} {31 \choose k}(16/30)^k(1-16/30)^{31-k} \approx 64.6\%
$$
This explicitly calculates the probability team A will win at least 16 games if all 31 games are played. In practice the teams will stop playing as soon as one team reaches 16 wins, but this does not change the resulting probability, because even if the match continues beyond that point team A's status of having either less than 16 wins or 16 or more wins cannot change.
The reason this seems high is because the teams are playing a lot of games. The more games the teams play, the closer team A's probability of winning the series will get to $100\%$. This is due to The Law of Large Numbers.
Update:
If teams A and B have played games in the same player pool, then you can use the Elo rating system to compute their ratings and estimate the probability of team A beating team B in a single game. This calculation will include the 30 head-to-head games the teams played as well as additional games against other opponents, and it will give you a more accurate and robust estimate of team A's win probability than the 16/30 you are using.
According to the Elo system, the probability of team A beating team B is:
$$
P(A\ beats\ B) = \frac{1}{1 + 10^{(R_B-R_A)/400}}
$$
Where $R_A$ is team A's rating, and $R_B$ is team B's rating.
