Answer to Question #115421 in Statistics and Probability for Emmanuel

Question

An archer shoots arrows at a circular target where the central portion of the target inside is called the bull. The archer hits the bull with probability 1/32. Assume that the archer shoots 96 arrows at the target, and that all shoots are independent
(a) Find the probability mass function of the number of bulls that the archer hits.
(b) Give an approximation for the probability of the archer hitting no more than one bull.

Answer 1

(a) Let X denote the number of shoots that hit the bull. Then X is binomially distributed:

P(X=k)=C_n,kp^k(1-p)^n-k=C_96,k(1/32)^k(31/32)^96-k ; n=96,p=1/32

(b) Since n is large, and p is small, we can use the Poisson approximation, with parameter λ=np=3 . Thus,

P(X≤1)=P(X=0)+P(X=1)=e^-λ +λ*e^-λ=4e^-3=0.199