Question

In the biathlon event of the Olympic Games, a participant skis cross-country and on four intermittent occasions stops at a rifle range and shoots a set of five shots. If the center of the target is hit, no penalty points are assessed. If a particular man has a history of hitting the center of the target with 86% of his shots, what is the probability of the following. (Give your answers correct to three decimal places.)

(a) He will hit the center of the target with all five of his next set of five shots.


(b) He will hit the center of the target with at least four of his next set of five shots. (Assume independence.)

Answer 1

In the biathlon event of the Olympic Games, a participant skis cross-country and on four intermittent occasions stops at a rifle range and shoots a set of five shots. If the center of the target is hit, no penalty points are assessed. If a particular man has a history of hitting the center of the target with 86% of his shots, what is the probability of the following. (Give your answers correct to three decimal places.)

1. He will hit the center of the target with all five of his next set of five shots.
   

Binomial distribution used. n=5, p=0.86

P(X=x) = (_nC_x) p^x (1-p)^n-x   

P(X=x) = (₅C₅) 0.86⁵ (1-0.86)^5-5   

P(X = 5) = 0.4704270176

=0.470 ( three decimals)

(b) He will hit the center of the target with at least four of his next set of five shots. (Assume independence.)

P( x ≥4) = P( x=4)+P( x=5)

= 0.470427+ 0.382906

=0.853333

=0.853 ( three decimals)