An airline company is considering a new policy of booking as many as 161 persons on an airplane that can seat only 140. (Past studies have revealed that only 81% of the booked passengers actually arrive for the flight.) Estimate the probability that if the company books 161 persons. not enough seats will be available.
To estimate the probability that there won't be enough seats available if the airline company books 161 persons, we need to consider the number of persons who actually show up for the flight.
Let's break down the problem step by step:
1. Calculate the expected number of persons who will show up for the flight:
Number of persons = 161
Probability of showing up = 81% = 0.81
Expected number of passengers showing up = Number of persons * Probability of showing up
Expected number of passengers showing up = 161 * 0.81 = 130.41 (rounding to the nearest integer)
2. Compare the expected number of passengers showing up with the number of available seats:
Number of available seats = 140
If the expected number of passengers showing up is greater than the number of available seats, there won't be enough seats for all the passengers.
3. Calculate the probability that there won't be enough seats available:
Probability = 1 - Cumulative Probability
Cumulative Probability can be calculated using the Binomial Distribution formula. Since the number of passengers showing up follows a binomial distribution with parameters (161, 0.81), we can use a calculator or software to find the cumulative probability.
Using an online calculator or statistical software, entering the parameters (161, 0.81), we find that the cumulative probability (P(X<=140)) is approximately 0.9321.
Therefore, the probability that there won't be enough seats available is:
Probability = 1 - 0.9321
Probability ≈ 0.0679 or 6.79%