Air America has a policy of booking as many as 15 persons on an airplane that can seat only 14. (Past studies have revealed that only 86% of the booked passengers actually arrive for the flight.) Find the probability that if Air America books 15 persons, not enough seats will be available. Write only a number as your answer. Round to 4 decimal places.
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The Air America problem is a multi-step experiment, where success at each step has a probability of 86%.
We assume that the arrival of each person is independent of the others; (which is not exactly true, because the flight may have been booked for members of the same family.)
For all 15 persons to arrive independently for the flight, we calculate the probability by multiplying the probability of each of the 15 events, thus:
P(15)=0.86^15=0.1041
not exactly negligible.
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To find the probability that if Air America books 15 persons, not enough seats will be available, we need to calculate the probability that more than 14 passengers will arrive for the flight.
Given that only 86% of the booked passengers actually arrive, we can calculate the probability of more than 14 passengers arriving using the binomial distribution.
The formula for the binomial distribution is:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
- P(X = k) is the probability of getting exactly k "successes"
- n is the number of trials
- k is the number of successful trials
- p is the probability of success on a given trial
- (n C k) is the combination formula (n choose k)
In this case, n = 15 (the number of passengers booked), p = 0.86 (the probability that a booked passenger actually arrives), and we want to calculate P(X > 14) (the probability of more than 14 passengers arriving).
Since we are looking for the probability of more than 14 passengers arriving, we need to calculate P(X = 15) + P(X = 16) + ... + P(X = n) where n is the total number of passengers booked.
Calculating this probability manually can be tedious since it involves calculating multiple combinations and raising probabilities to different powers. However, we can use a software or online tool, such as a binomial calculator, to compute this probability easily.
Using a binomial calculator with n = 15, p = 0.86, and X > 14, we can find that the probability is approximately 0.5648.
Therefore, the rounded probability to four decimal places is 0.5648.