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.
To find the probability that not enough seats will be available when Air America books 15 persons, we need to consider the probability that more than 14 passengers will arrive for the flight.
Given that only 86% of the booked passengers actually arrive for the flight, we can calculate the probability that exactly 14 passengers will arrive. This can be done using the binomial probability formula.
The binomial probability formula is:
P(X = k) = nCk * p^k * q^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes,
- n is the total number of trials,
- k is the number of successes,
- p is the probability of success on a single trial,
- q is the probability of failure on a single trial (q = 1 - p), and
- nCk represents the number of possible combinations of choosing k successes from n trials (nCk = n! / (k!(n-k)!)).
In this case, n = 15 (since Air America books 15 persons), k = 14 (we are interested in the case where 14 passengers arrive), and p = 0.86 (probability of a passenger showing up).
Calculating nCk:
nCk = n! / (k!(n-k)!) = 15! / (14!(15-14)!) = 15
Substituting the values into the formula:
P(X = 14) = 15 * (0.86^14) * (0.14^1)
Now, to find the probability that more than 14 passengers will arrive, we sum up the probabilities of getting 15 passengers:
P(X > 14) = 1 - P(X = 14)
Calculating P(X > 14):
P(X > 14) = 1 - P(X = 14) = 1 - (15 * (0.86^14) * (0.14^1))
Now, we can calculate the probability that not enough seats will be available when Air America books 15 persons by rounding the result to 4 decimal places.