The problem goes like this:
Alice and Bob are playing a game that they cannot directly observe. The game starts with a referee rolling a ball on a billiard table and marking the location where it comes to rest. Once the initial location is marked, the referee rolls one ball at a time and awards one point to Alice if the ball lands to the left of the mark, and awards one point to Bob if the new ball lands to the right of the mark. After eight rounds Alice has five points and Bob has three points. The first person to six points wins the game. For the sake of simplicity, we will assume that the referee is unbiased, the table is fair, and the balls have a uniform probability of landing anywhere on the table.
At this point, what is the probability that Bob will win the game?
Using the Bayesian method and the Monte Carlo method, the answer is around 0.09. Using the "naive" frequentist method, the answer is around 0.05, which is incorrect. I've often heard of a "non-naive" frequentist method which gives the correct answer eluded to, but I can't find one and don't know how it would go.
How can frequentist statistics be used to solve this problem correctly?
Edit: According to the "naive" frequentist view, the chance that Bob gets a point is equal to 3/8, so the probability that he gets three points in a row to win the game is (3/8)^3, or around 0.05.
