The probability that an amateur archer hits a target 30 feet

The probability that an amateur archer hits a target 30 feet away is 20%. Suppose we take a sample of 12 archers.

            a. What is the expected number of archers that will hit the target?

            b. What is the standard deviation?

            c. What is the probability that exactly 3 archers hit the target?

            d. What is the probability that more than 4 archers hit the target?

Solution

a)
expected number of archers ( np ) = 12 * 0.20 = 2.4
b)
Standard Deviation ( npq )= 12*0.2*0.8 = 1.3856
c)
P( X = 3 ) = ( 12 3 ) * ( 0.2^3) * ( 1 - 0.2 )^9
= 0.2362
          
d)
P( X < 4) = P(X=3) + P(X=2) + P(X=1) + P(X=0) +
= ( 12 3 ) * 0.2^3 * ( 1- 0.2 ) ^9 + ( 12 2 ) * 0.2^2 * ( 1- 0.2 ) ^10 + ( 12 1 ) * 0.2^1 * ( 1- 0.2 ) ^11 + ( 12 0 ) * 0.2^0 * ( 1- 0.2 ) ^12 +   
= 0.7946

P( X > = 4 ) = 1 - P( X < 4) = 0.2054