Answer to Question #115549 in Statistics and Probability for Albert

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) $P(x)=C_{96}^x (\frac{1}{32})^x(\frac{31}{32})^{96-x}$

(b) Using the Poisson approximation with $\lambda=np=96*\frac{1}{32}=3.$

$P(X\le1)=P(X=0)+P(X=1)=e^{-\lambda}+\lambda e^{-\lambda}=e^{-3}+3e^{-3}=0.1991.$