An archer hits a bulls eye with a probability of 009 Suppose

An archer hits a bull’s eye with a probability of 0.09. Suppose that the archer misses the target completely with a probability of 0.12. If the archer shoots eight arrows, calculate the following using 5 decimal places:

(a) probability of scoring at least two bull’s eyes

(b) probability of scoring exactly one bull’s eye and missing the target exactly twice

(c) expected number of times that the archer misses the target completely

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 < 2) = P(X=1) + P(X=0)   
= ( 8 1 ) * 0.09^1 * ( 1- 0.09 ) ^7 + ( 8 0 ) * 0.09^0 * ( 1- 0.09 ) ^8
= 0.84232
P( X > = 2 ) = 1 - P( X < 2) = 0.15768

c)
Mean ( np ) = 8 * 0.12 = 0.96
Standard Deviation ( npq )= 8*0.12*0.88 = 0.9191
Normal Distribution = Z= X- u / sd