A and B throw alternatively a pair of dice, A wins if he throws a sum

Solution:

Recall that there are 36 possible outcomes when a pair of dice is thrown. The outcomes yielding a sum of 8 are: (2,6), (3,5), (4,4), (5,3), and (6, 2). Thus, the probability that A wins on any given throw is 5/36, and the probability that A doesn’t win is 31/36.

For B, there are 4 outcomes yielding a sum of 9: (3, 6), (4, 5), (5, 4), and (6, 3), and so the probability that B wins on any given throw is 4/36 = 1/9. The probability that B doesn’t win on any given throw is 8/9.

The experiment describes a geometric random variable, in which the game stops when A rolls a sum of 8 (and B has not yet won). Let’s determine the first several trials of this experiment to understand the pattern that develops:

First throw:
P(A wins on first throw) = 5/36

Second throw:
P(A wins on second throw) = P(A wins on second throw) x P(A didn’t win on first throw and B didn’t win on first throw)
P(A wins on second throw) = 5/36 x [31/36 x 8/9]

Third throw:
P(A wins on third throw) = P(A wins on third roll) x P(A didn’t win on first throw and B didn’t win on first throw) x P(A didn’t win on second throw and B didn’t win on second throw)
P(A wins on third throw) = 5/36 x [31/36 x 8/9] x [31/36 x 8/9]
P(A wins on third throw) = 5/36 x [(31/36)^2 x (8/9)^2]

We want to answer the question of calculating the probability that A wins the game. This happens if A wins on the first throw OR A wins on the second throw OR A wins on the third throw, or fourth, or fifth, etc. Because the number of throws needed until A wins is unknown, let’s sum the probabilities already calculated and extend the sum infinitely.

P (A wins game) = 5/36 + {5/36 x [31/36 x 8/9]} + {5/36 x [31/36 x 8/9] x [31/36 x 8/9]} + …

We can see that we are generating the sum of a geometric series, with first term a = 5/36 and multiplier r = [31/36 x 8/9]. The formula for the sum of a geometric series a / (1 – r) can be used to calculate the probability that A wins the game:

a / (1 – r) = 5/36 / [1 – 31/36 x 8/9] = 5/36 / (1 – 248/324) = (5/36) / (76/324) = 45/76

Answer: B