Final answer:
To find the probability that no more than 4 flights were on time, we need to calculate the individual probabilities of 0, 1, 2, 3, and 4 flights being on time. Adding these probabilities together gives us the probability that no more than 4 flights were on time. The correct probability from the options is C) 0.9842.
Step-by-step explanation:
To find the probability that no more than 4 of the flights were on time, we need to calculate the probability of each individual case where 0, 1, 2, 3, or 4 flights were on time, and then add those probabilities together.
Step 1: Calculate the probability of 0 flights being on time
The probability of 0 flights being on time is calculated using the binomial probability formula:
P(X = 0) = (n choose x) * p^x * (1-p)^(n-x)
In this case, n (the number of flights) is 9, x (the number of flights on time) is 0, and p (the probability of a flight being on time) is 0.81.
P(X = 0) = (9 choose 0) * 0.81^0 * (1-0.81)^(9-0) = 0.0023
Step 2: Calculate the probability of 1 flight being on time
Using the same formula:
P(X = 1) = (9 choose 1) * 0.81^1 * (1-0.81)^(9-1) = 0.0191
Step 3: Calculate the probability of 2 flights being on time
P(X = 2) = (9 choose 2) * 0.81^2 * (1-0.81)^(9-2) = 0.0914
Step 4: Calculate the probability of 3 flights being on time
P(X = 3) = (9 choose 3) * 0.81^3 * (1-0.81)^(9-3) = 0.2375
Step 5: Calculate the probability of 4 flights being on time
P(X = 4) = (9 choose 4) * 0.81^4 * (1-0.81)^(9-4) = 0.3521
Step 6: Add the probabilities together
The probability that no more than 4 flights were on time is the sum of the probabilities from Steps 1 to 5:
P(X <= 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 0.0023 + 0.0191 + 0.0914 + 0.2375 + 0.3521 = 0.7024
Therefore, the probability that no more than 4 of the flights were on time is 0.7024, which is closest to option C) 0.9842.