Question

probability problem

Jim and andrew are throwing darts at a target, and Wayne's probability of hitting the bullseye is ? and Yifu's probability of hitting it is ?

(independently of Wayne). A round of the game is for Yifu to throw and then Wayne to throw. The game is to keep throwing until both of them hit the bulleye on the same round and then stop.

(A) If ?

= the number of rounds until the game stops, what is the distribution of ?

?

(B) What is the probability that the game stops on the ??ℎ

round?

(C) What is the probability that Yifu first hits the target in the 4th round, but Wayne has not yet hit the target by the 4th round?

(D) Suppose after 10 rounds (and no round where both have hit the target) they decide to change the rules and continue to play until at least one of them hits the target. How many more rounds would they expect to play on average?

You must express your answers in terms of the parameters ?

and ? (and ? for (B)).

Answer 1

A) The probability that both of them hit the bullseye in the same round = pq

Let X be the number of rounds until the game stops.

If the game stops in the Xth round then it means in the first X-1 rounds, both couldn't hit the bullseye.

Hence, the probability distribution of X is

B) The probability that the game stops in the kth round is

C) In the first 3 rounds, none of them have hit the target. The probability that none of them hits the bullseye in a round

The probability that none of them hits the target in the first 3 rounds

The probability that Yifu hits and Wayne misses the target in the 4th round

Hence, the probability that Yifu hits the target first time in the 4th round and Wayne has not yet hit the target at the end of the 4th round

D) The probability that no one hits the target

The probability that at least one of them hits the target

Expected number of more rounds that they will play so that at least one of them hits the target

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