The rates of ontime flights for commercial jets are continuo

The rates of on-time flights for commercial jets are continuously tracked by the U.S. Department of Transportation. Recently, Southwest Air had the best reate with 80 % of its flights arriving on time. A test is conducted by randomly selecting 17 Southwest flights and observing whether they arrive on time.

(a) Find the probability that at least 4 flights arrive on time.

(b) Would it be unusual for Southwest to have 15 flights arrive on time?
(Enter YES or NO)

Solution

Binomial distribution with p = 0.80 and n = 17
P(x) = nCx * (p^x) * ((1-p)^(n-x))

a.
probability that at least 4 flights arrive on time
= P(X>=4)
= 1 - (P(0) + P(1) + P(2) + P(3))
= 1 - ((17C0 * 0.80^0 * 0.20^17) + (17C1 * 0.80^1 * 0.20^16) + (17C2 * 0.80^2 * 0.20^15) + (17C3 * 0.80^3 * 0.20^14))
= 1 - ((1 * 0.80^0 * 0.20^17) + (17 * 0.80^1 * 0.20^16) + (136 * 0.80^2 * 0.20^15) + (680 * 0.80^3 * 0.20^14))
= 0.99999994

b.
P(15) = 17C15 * 0.80^15 * 0.20^2
= 17C15 * 0.80^15 * 0.20^2
= 136 * 0.80^15 * 0.20^2
= 0.1914

Since probability is greater than 0.05 hence it would NOT be unusual for Southwest to have 15 flights arrive on time.