Airline Reservations In Chapter 6, we learned that the proportion of passengers who miss a flight for which they have a reservation is 0.0995. Suppose a flight has 320 reservations, but only 300 seats on the plane. What is the probability that 300 or fewer passengers show up for the flight?

Verified step by step guidance

1. Identify the random variable of interest: Let X be the number of passengers who show up for the flight out of the 320 reservations.
2. Recognize that X follows a binomial distribution with parameters n = 320 (number of trials) and p = 1 - 0.0995 = 0.9005 (probability a passenger shows up). So, \(X \sim \text{Binomial}(n=320, p=0.9005)\).
3. We want to find the probability that 300 or fewer passengers show up, which is \(P(X \leq 300)\).
4. Since n is large, approximate the binomial distribution with a normal distribution using the normal approximation to the binomial. Calculate the mean and standard deviation of X: \(\mu = np\) and \(\sigma = \sqrt{np(1-p)}\).
5. Apply the continuity correction by finding \(P(X \leq 300.5)\) in the normal distribution, then convert this to a standard normal variable \(Z = \frac{300.5 - \mu}{\sigma}\) and use standard normal tables or software to find the corresponding probability.

Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. In this problem, each passenger either shows up (success) or misses the flight (failure), with a known probability. This distribution helps calculate the probability of a certain number of passengers showing up.

Normal Approximation to the Binomial

When the number of trials is large, the binomial distribution can be approximated by a normal distribution to simplify calculations. This involves using the mean and standard deviation of the binomial distribution and applying a continuity correction. This approximation is useful here due to the large number of reservations (320).

Probability of Events and Cumulative Probability

Calculating the probability that 300 or fewer passengers show up involves finding the cumulative probability up to 300. This means summing the probabilities of all outcomes where the number of passengers showing up is 300 or less. Understanding cumulative probability is essential to answer the question accurately.