help with stats hw

Question

According to an​ airline, flights on a certain route are on time

75​% of the time. Suppose 24

flights are randomly selected and the number of​ on-time flights is recorded.

​(a) Explain why this is a binomial experiment.

​(b) Find and interpret the probability that exactly

14 flights are on time.

​(c) Find and interpret the probability that fewer than 14

flights are on time.

​(d) Find and interpret the probability that at least

14 flights are on time.

​(e) Find and interpret the probability that between

12 and 14 flights, inclusive, are on time.

Answer 1

a) Each instance is a success (on time) or a failure (not on time). The number of successes goes according to a binomial distribution.

b) Let X = # successes in 24 trials. P(X=14) = (24 choose 14) * .75¹⁴ * .25¹⁰ ≈ .0333

In words, this says

(1) That the probability of any sequence with 14 successes and 10 failures is Prob(success)¹⁴ * Prob(failure)¹⁰, and

(2) According to the binomial theorem, there are (24 choose 14) such sequences

c) P(X < 14) is the sum of P(X=0) + P(X=1) + ... + P(X=13) ≈ .0213

where each of the terms in the sum is computed in similar fashion to what we did for part b)

d) P(X ≥ 14) = 1 - P(X < 14) ≈ 0.979

e) P(12 ≤ X ≤ 14) = P(X = 12) + P(X = 13) + P(X = 14) ≈ .0526

The probabilities P(X=14) and P(X<14) are small because the probability of an individual flight being on time is much greater than 1/2. This exercise quantifies how much the distribution is "skewed to the right".