Question

A Naval intercept squadron consists of 16 planes that should always be ready for immediate launch. However, a plane’s engines are troublesome, and there is a probability of 0.25 that the engines of a particular plane will not start at a given attempt. The squadron commander is interested in how many planes will immediately become airborne if the squadron is called to action.

a) What is the probability that at least 10 aircraft will be ready to be airborne at any given time?

b) What is the expected number of planes that will be ready to launch on time?

c) What is the standard deviation?

Answer 1

This is a binomial distribution.

p = 1 - 0.25 = 0.75

n = 16

P(X = x) = 16Cx * 0.75^x * (1 - 0.75)^16-x

a) P(X > 10) = P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15) + P(X = 16)

                    = 16C10 * 0.75¹⁰ * 0.25⁶ + 16C11 * 0.75¹¹ * 0.25⁵ + 16C12 * 0.75¹² * 0.25⁴ + 16C13 * 0.75¹³ * 0.25³ + 16C14 * 0.75¹⁴ * 0.25² + 16C15 * 0.75¹⁵ * 0.25¹ + 16C16 * 0.75¹⁶ * 0.25⁰

                    = 0.9204

b) Expected value = n * p = 16 * 0.75 = 12

c) standard deviation = sqrt(n * p * (1 - p)) = sqrt(16 * 0.75 * 0.25) = 1.732