Question

The Problem of the Points. Two players engage in a game. They are equally matched, and the chances of winning a point are the same for each player. The first player to score 10 points wins the $100 stakes. The game is interrupted suddenly when Player A has 7 points and Player B has 5 points. How should the stakes be divided fairly

Answer 1

Player A receives $77.34375 of the sakes

Player B receives $22.65625 of the stakes

Explanation:

Given that at the time the game was interrupted, we have;

The number of points player A has = 7 points

The number of points player B has = 5 points

The amount the first player to score 10 points wins = $100

The number of points remaining for player A to win the game = 3

The number of points remaining for player B to win the game = 5

Therefore, the number of trials remaining for a winner to emerge = 3 + 5 - 1 = 7 trials

We take the probability that Player A wins a point as success

We find the likelihood of Plater A winning by using binomial theory as follows;

`P(S) = \sum\limits_(j=3)^7 \dbinom{7}{3} \cdot (1)/(2^7) = (35)/(128) + (35)/(128) + (21)/(128) + (7)/(128) + (1)/(128) = (99)/(128)`

Therefore, given that the likelihood of Player A winning = 99/128

The stakes should be divided such that Player A gets 99/128 share of the stakes while Player B gets 1 - 99/128 = 29/128 share of the stakes

Therefore, Player A gets (99/128) × $100 = $77.34375

Player B gets (29/128) × $100 = $22.65625