The probability that an amateur archer hits a target 50 feet

The probability that an amateur archer hits a target 50 feet away is 20%. Suppose we take a sample of 20 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 7 archers hit the target?

            d. What is the probability that more than 2 archers hit the target

Solution

The probability that an amateur archer hits a target 50 feet away is 20%. Suppose we take a sample of 20 archers.

Binomial distribution used

            n=20

p=0.20

Expectation = np = 4

expected number of archers that will hit the target =4

Variance = np(1 - p) = 3.2

Standard deviation = 1.7889

P(X=x) = (_nC_x) p^x (1-p)^n-x  

P(X=7) = (₂₀C₇) 0.2⁷ (1-0.2)^20-7  

P(X = 7) = 0.0545

            d. What is the probability that more than 2 archers hit the target

p(x >2) = 1 - ( p(x=0)+p(x=1)+p(x=2))

= 1- ( 0.0115+ 0.0576+ 0.1369) =0.794

The required    P=0.794