An airline has a policy of booking as many as 22 persons on an airplane that can seat only 21. ( past studies have revealed that only 94% of the booked passengers actually arrive for the flight. ) find the probability that if the airline books 22 persons, not enough seats will be available
To find the probability that not enough seats will be available when the airline books 22 persons, we need to consider two factors:
1. The probability that more than 21 passengers will actually arrive.
2. The probability that exactly 21 passengers will arrive.
First, let's calculate the probability that more than 21 passengers will arrive. Since the past studies reveal that only 94% of the booked passengers actually arrive, the probability of any individual passenger arriving is 0.94 (or 94%). The probability that more than 21 passengers will arrive can be calculated using the binomial probability formula:
P(X > 21) = 1 - [P(X = 0) + P(X = 1) + ... + P(X = 21)]
Where X is a random variable representing the number of passengers arriving.
Using this formula, we can calculate the probability of 0, 1, 2, ..., 21 passengers arriving, and subtract it from 1 to get the probability of more than 21 passengers arriving.
Next, let's calculate the probability that exactly 21 passengers will arrive. This can be calculated using the binomial probability formula as well:
P(X = 21) = (nCk) * (p^k) * ((1-p)^(n-k))
Where n is the number of trials, k is the number of successes, and p is the probability of success.
In this case, n = 22 (number of booked passengers), k = 21 (number of seats available), and p = 0.94 (probability of an individual passenger arriving).
By calculating both the probability of more than 21 passengers and the probability of exactly 21 passengers arriving, you can then add them together to find the overall probability that not enough seats will be available.