The probability is p that Marty hits target M when he fires

The probability is p that Marty hits target M when he fires at it. The probability is q that Alvie hits target A when he fires at it. Marty and Alvie fire one shot each at their targets. If both of them hit their targets. they stop; otherwise. they will continue. (a) What is the probability that they stop after each has fired r times? (b) What is the expected value of the number of the times that each of them has fired before stoping?

Solution

The probability that they both hit their targets (and thus stop) is pq.
This problem follows the geometric distribution with probability = pq.
Then, the probability of stopping on the r\'th time is simply (1-pq)^r-1 pq

(You can easily derive this formula; you do not hit the target with probability 1 - pq, so to both hit the target for the first time on the r\'th try, you failed r-1 times and then succeeded on the r\'th time, so the probability is (1-pq)^r-1 pq as stated above.

The mean number of times until success is 1/probability of success = 1/pq

This is true for the geometric distribution. r(1-pq)^r-1 pq = 1/pq

See http://en.wikipedia.org/wiki/Geometric_distribution for a derivation.