Ann and Bill play rock-paper-scissors. Each has a strategy of choosing uniformly at random out of the three possibilities every round (independently of the other player and the previous choices). (a) What is the probability that Ann wins the first round? (Remember that the round could end in a tie.) (b) What is the probability that Ann's first win happens in the fourth round? (c) What is the probability that Ann's first win comes after the fourth round?

a) Ann wins in the first round with probability 1/3

b) The probability that Ann's first win is in the fourth round in 8/81

c) The probability that Ann's first win comes after the fourth round is 16/81

Step-by-step explanation:

a) Each strategy is played with a probability of 1/3. Given each strategy, there is a probability of 1/3 that Bill plays the strategy that makes Ann the winner. Therefore, the probability that Ann wins in the first raound (and, in fact, in any round) is

1/3 * 1/3 + 1/3 * 1/3 + 1/3 * 1/3 = 1/9 + 1/9 + 1/9 = 1/3.

The probability that Ann wins in the first round is 1/3.

b) Ann winning in a round happens with probaiblity 1/3, therefore, she wont win in a round with probability 2/3. This should happen 3 times before the first win, thus, the probability that ann's first win happens in the fourth round is

(2/3) ³ * 1/3 = 8/81

c) The first win comes after the fourth found if she doesnt win in the first 4 times, and this will happen with a probability of (2/3) ⁴ = 16/81