An airline estimates that 91% of people booked on their flights actually show up. If the airline books 80 people on a flight for which the maximum number is 78, what is the probability that the number of people who show up will exceed the capacity of the plane?
To find the probability that the number of people who show up will exceed the capacity of the plane, we need to find the probability of booking more than 78 people given that 91% of people booked actually show up.
Let's calculate the expected number of people who will show up on the flight:
Expected number of people = 91% of the booked number
Expected number of people = 91% of 80
Expected number of people = 0.91 * 80
Expected number of people = 72.8
So, the expected number of people who will show up is 72.8.
Now, let's calculate the probability of booking more than 78 people:
Probability (booking more than 78 people) = 1 - Probability (booking 78 or fewer people)
To find the probability of booking 78 or fewer people, we can use a binomial distribution with n = 80 (number of bookings) and p = 0.91 (probability of showing up). We need to sum the probabilities of booking 0, 1, 2, ..., 78 people.
Using a calculator or software, we can calculate the cumulative probability as follows:
P(X ≤ 78) = ∑ (i=0 to 78) (nCi * p^i * q^(n-i))
Where nCi represents the combination formula "n choose i" and q = 1 - p.
Once we have the probability of booking 78 or fewer people, we can subtract it from 1 to find the probability of booking more than 78 people.
Alternatively, using statistical software or online calculators, you can directly calculate the probability of booking more than 78 people by specifying the parameters of the binomial distribution.
So, in summary, to find the probability that the number of people who show up will exceed the capacity of the plane, you'll need to calculate the probability of booking 78 or fewer people and subtract it from 1.