This is all one question In the biathlon event of the Olympi

This is all one 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 91% 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.)

Solution

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

a)
P( X = 5 ) = ( 5 5 ) * ( 0.91^5) * ( 1 - 0.91 )^0
= 0.62403
b)

P( X < 4) = P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 5 3 ) * 0.91^3 * ( 1- 0.91 ) ^2 + ( 5 2 ) * 0.91^2 * ( 1- 0.91 ) ^3 + ( 5 1 ) * 0.91^1 * ( 1- 0.91 ) ^4 + ( 5 0 ) * 0.91^0 * ( 1- 0.91 ) ^5
= 0.0673
P( X > = 4 ) = 1 - P( X < 4) = 0.9326