The following famous problem is centuries old. Over the 16 th and 17 th centurie

Question

The following famous problem is centuries old. Over the 16th and 17th centuries it prompted several solutions and much debate. Try to determine a fair numerical resolution to the problem. The original formulation of the problem appears below:

A and B are playing a fair game of balla. They agree to continue until one has won six rounds. The game actually stops when A has won five and B four. How should the stakes be divided?

In other words, each player has an equal chance of winning any given round. The first person to win six rounds gets all of the winnings in the pot. The game stops prematurely when A has won five rounds and B has won four rounds. How should the winnings be fairly divided at this point?

Explanation / Answer

B has won 4 games. Had the game continued, the only way B would have won was by winning 2 consecutive games. The probability that wins the next game is 1/2 and the probability that he wins the game after that is also 1/2. So the probability that B wins both games is 1/2*1/2 = 1/4.

Since B's probability is 1/4, A's is 3/4.

So A should get 3/4 th of the stake.

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