Two players are playing a game at which they are equally skilled. The winner is

Question

Two players are playing a game at which they are equally skilled. The winner is to be the first who wins 7 rounds. Unfornuately, the game is interrupted when player A has won 5 rounds and player B has won 3 rounds. How should the stakes be split? More generally, if the probabilty that player A wins a round is p and the probability that player A wins a round is q (s.t. q=1-p), the winner is the first to win n rounds. A has already won x rounds and B has won y rounds, how should the stakes be allocated?

Explanation / Answer

Probability of winning by A=Probability of winning 2 games in future (2,3,4or5 games) = 1/2^2 + (3-1)*1/2*1/2^2+ (6-2)*1/2^2*1/2^2+(10-3)*1/2^2*1/2^3= 0.96875

[First 1/2 corresponds to p and second 1/2 corresponds to q]

A should be given 96.875% of the total prize.

Let x>y

Generalizing = Probability of winning for a=Probability of winning (n-x) games in (2n-(x+y)+1 rounds)

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