Final answer:
To find the probability of specific numbers of on-time flights, we can use the binomial probability formula. For exactly 3 flights, the probability is 0.0578. For at most 3 flights, the probability is 0.0426. For at least 3 flights, the probability is 0.8785.
Step-by-step explanation:
a. To find the probability that exactly 3 flights are on time, we can use the binomial probability formula. The formula is P(X=k) = C(n, k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, p is the probability of success, and C(n, k) is the number of combinations. In this case, n = 8, k = 3, and p = 0.7. Plugging in these values, we get P(X=3) = C(8, 3) * 0.7^3 * (1-0.7)^(8-3) = 0.0578.
b. To find the probability that at most 3 flights are on time, we need to find the probabilities of 0, 1, 2, and 3 flights being on time and then add them together. Using the same binomial probability formula, we can calculate P(X≤3) = P(X=0) + P(X=1) + P(X=2) + P(X=3) = C(8, 0) * 0.7^0 * (1-0.7)^(8-0) + C(8, 1) * 0.7^1 * (1-0.7)^(8-1) + C(8, 2) * 0.7^2 * (1-0.7)^(8-2) + C(8, 3) * 0.7^3 * (1-0.7)^(8-3) = 0.0426.
c. To find the probability that at least 3 flights are on time, we need to find the probabilities of 3, 4, 5, 6, 7, and 8 flights being on time and then add them together. Using the same binomial probability formula, we can calculate P(X≥3) = P(X=3) + P(X=4) + P(X=5) + P(X=6) + P(X=7) + P(X=8) = 0.8785.