Ann and Bill play rock-paper-scissors. Each has a strategy of choosing uniformly at random out of the three possibilities every

Question

3 months ago

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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?

Mathematics

1 answer:

Inessa [12.5K] · Answer 1 · 2026-03-04T16:34:14+03:00

Answer:

a) Ann has a 1/3 chance of winning in the first round

b) The chance of Ann winning for the first time in the fourth round is 8/81

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

Step-by-step explanation:

a) Each strategy is played with a probability of 1/3. Given any strategy, there’s a 1/3 chance that Bill will choose the strategy that allows Ann to win. Consequently, the probability of Ann securing a victory in the first round (or any round) is

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

Thus, the likelihood of Ann winning the initial round is 1/3.

b) The chances of Ann winning a round stand at 1/3; therefore, her chances of not winning are 2/3. This must happen three times before her first victory. Thus, the probability that Ann's first win occurs in the fourth round is

(2/3)³ * 1/3 = 8/81.

c) The first victory happens after the fourth round if she remains unsuccessful in the first four rounds, translating to a possibility of (2/3)⁴ = 16/81.