A and B shoot independently until each shoots their target. They have probabilities 3 5 and 5 7 respectively of hitting the target at each shot. Then:

Step-by-step solution

Let a and b be the number of shots A, B respectively takes until they hit the target. S= _ n=1 ^ (P(a=n)) (P(b>n)) Now, for P(a=n), A must miss their first n-1 shots and hit on the n th shot. Thus, probability of A=n= (1- 3 5 )^ n-1 3 5 = 3 2 ( 2 5 )^n Next we have P(b>n)= _ k=n+1 ^ P(b=k)= _ k=n+1 ^ ( 2 7 )^ k-1 5 7 = ( 2 7 )^n Thus, S= 3 2 _ n=1 ^ ( 2 5 2 7 )^n= 3 2 4 35 1- 4 35 = 6 31 (1) Now, probability that A, B require the same number of shots will be _ n=1 ^ (P(A=n)) (P(B=n))= _ n=1 ^ ( ( 4 35 )^ n-1 3 5 5