About 24% of flights departing from New York's John F. Kennedy International Airport were delayed in 2009. Assuming that the chance of a flight being delayed has stayed constant at 24%, we are interested in finding the probability of 10 out of the next 100 departing flights being delayed. Noting that if one flight is delayed, the next flight is more likely to be delayed, which of the following statements is correct?

Question

A. We can use the geometric distribution with n = 100, k = 10, and p = 0.24 to calculate this probability.
B. We can use the binomial distribution with n = 100, k = 10, and p = 0.24 to calculate this probability.
C. We can use the binomial distribution with n = 10, k = 100, and p = 0.24 to calculate this probability.
D. We cannot calculate this probability using the binomial distribution since whether or not one flight is delayed is not independent of another.

Answer 1

To analyze the situation, we note that the question involves calculating the probability of a certain number of delays occurring among a set of flights, given a fixed probability of delay for each flight. The flight delay data suggests that the events can be modeled using a statistical distribution.

1. Independence of Events: The assumption of independent events is central to determining the appropriate distribution to use. While the statement mentions that if one flight is delayed, the next flight is more likely to be delayed, this indicates a dependence between events. However, in many practical applications, especially in cases with large sample sizes where probabilities are relatively small (like 24%), it's common to use the binomial distribution as an approximation under the assumption of independence.

2. Choice Analysis:

   • Option A: Incorrect. The geometric distribution is used to model the number of trials until the first success, not for counting successes in a fixed number of trials.
   • Option B: Correct. The binomial distribution can be used to calculate the probability of exactly 10 delays out of 100 flights, given that the probability of delay is 0.24 (n=100, k=10, p=0.24).
   • Option C: Incorrect. In this case, n should represent the number of trials (flights), and k the number of successes (delays). The values are incorrectly assigned.
   • Option D: While the dependencies mentioned suggest complications in using the binomial model, it's still common to apply the binomial distribution for practicality when dealing with a large number of trials and a relatively stable probability.

Given this analysis, the most suitable and correct choice is:

B. We can use the binomial distribution with n = 100, k = 10, and p = 0.24 to calculate this probability.