Question

According to an airline, flights on a certain route are on time 80% of the time. Suppose 15 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 9 flights are on time.
(c) Find and interpret the probability that fewer than 9 flights are on time.
(d) Find and interpret the probability that at least 9 flights are on time.

Answer 1

The student's question is a binomial experiment involving 15 flights with 80% chance of being on time. To find probabilities for specific outcomes, the binomial probability formula is used with combinations and respective powers of success and failure probabilities.

Step-by-step explanation:

The question involves a binomial experiment because it adheres to the criteria of having two possible outcomes (on time or not on time), a fixed number of trials (15 flights), and each trial being independent with the probability of success (flight being on time) remaining constant (80%).

To find the probability that exactly 9 flights are on time, we use the binomial probability formula:

1. Identify the variables: here, n=15 (number of flights), x=9 (number of on-time flights we're looking for), p=0.8 (probability of a flight being on time), q=0.2 (1 - p, probability of a flight being late).
2. Apply the formula: P(X=x) = C(n, x) * p^x * q^(n-x), where C(n, x) is the combination of n items taken x at a time.
3. Using a calculator or software to find P(X=9).

The probability that fewer than 9 flights are on time can be calculated by summing the probabilities for 0 to 8 on-time flights (P(X<9) = P(X=0) + P(X=1) + ... + P(X=8)).

For the probability that at least 9 flights are on time, we need to calculate 1 minus the probability of having fewer than 9 flights on time (P(X≥9) = 1 - P(X<9)).

Through calculations using the binomial formula, you can find the exact probabilities required for each part of the question.