Question

wo people, A and B, agree to play a series of fair games until one person has won five games. They each have wagered the same amount of money, the intention being that the winner will be awarded the entire pot. But suppose, for whatever reason, the series is prematurely terminated, at which point A has won four games and B three. Person A should receive what percentage of the stake?

Answer 1

The first 7 games are finished, A leading 4 wins to 3. If A wins game 8 (50% chance), then A leads 5-3 and the match ends. So, half of the times, A wins.

In the remaining half of the times (i.e. if B wins game 8, for a score of 4-4), the game is even: either A or B will win game 9 and make the final point.

So, 1/2 of the times A wins, and in the remaining 1/2 of the times, A wins with probability 1/2 and so does B.

This means that A wins with probability

$\dfrac{1}{2}+\dfrac{1}{2}\dfrac{1}{2}=\dfrac{3}{4}$

And B wins with probability

$\dfrac{1}{2}\dfrac{1}{2}=\dfrac{1}{4}$

So, A should take 3/4 of the pot.