Question

your flight has been delayed. at Philadelphia international airport, 80% of recent flights have arrived on time. a sample of 14 flights is studied. a) find the probability that exactly 14 of the flights were on time. b) find the probability that at least 12 of the flights were on time.c) find the probability that at most 10 of the flights were on time. d) would it be unusual for less than 9 of the flights to be on time.e) find the mean. f) find the standard deviation.

Answer 1

p = 0.8

n = 14

It is a binomial distribution.

P(X = x) = nCx * p^x * (1 - p)^{n - x}

a) P(X = 14) = 14C14 * (0.8)^14 * (0.2)^0 = 0.0440

b) P(X > 12) = P(X = 12) + P(X = 13) + P(X = 14)

                    = 14C12 * (0.8)^12 * (0.2)^2 + 14C13 * (0.8)^13 * (0.2)^1 + 14C14 * (0.8)^14 * (0.2)^0

                    = 0.4481

c) P(X < 10) = 1 - (X > 10)

                    = 1 - (P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14))

                    = 1 - (14C11 * (0.8)^11 * (0.2)^3 + 14C12 * (0.8)^12 * (0.2)^2 + 14C13 * (0.8)^13 * (0.2)^1 + 14C14 * (0.8)^14 * (0.2)^0)

                    = 1 - 0.6982 = 0.3018

d) P(X < 9) = 1 - P(X > 9)

                  = 1 - (14C9 * (0.8)^9 * (0.2)^5 + 14C10 * (0.8)^10 * (0.2)^4 + 14C11 * (0.8)^11 * (0.2)^3 + 14C12 * (0.8)^12 * (0.2)^2 + 14C13 * (0.8)^13 * (0.2)^1 + 14C14 * (0.8)^14 * (0.2)^0)

                 = 1 - 0.9561 = 0.0439

e) mean = np = 14 * 0.8 = 11.2

f) Standard deviation = sqrt(np(1 - p))

                                  = sqrt(14 * 0.8 * 0.2) = 1.497