An airline experiences a no-show rate of 6%. What is the maximum number of reservations that it could accept for a flight with a capacity of 160, if it wants the probability of accommodating all reservation holders to be at least 95%?
To calculate the maximum number of reservations that the airline could accept, taking into account the no-show rate and the desired probability of accommodating all reservation holders, we can use the concept of binomial probability.
Binomial probability calculates the probability of a certain number of successes (in this case, passengers showing up) in a series of independent events (reservations). The formula for binomial probability is:
P(x) = C(n, x) * p^x * (1 - p)^(n - x)
Where:
- P(x) is the probability of x successes
- C(n, x) is the number of combinations to choose x out of n reservations (calculated using the formula n! / (x! * (n-x)!))
- p is the probability of success on a single event (passenger showing up)
- (1 - p) is the probability of failure on a single event (passenger not showing up)
- n is the total number of events (reservations)
In this case, the airline wants the probability of accommodating all reservation holders to be at least 95%, which means the probability of at least x successes should be at least 95%.
Let's calculate the maximum number of reservations the airline can accept for a flight with a capacity of 160:
We need to find the largest value of x for which P(x) is greater than or equal to 95%. We can start by trying different values of x until we find the maximum:
When x = 160 (all passengers show up), the probability is P(160) = C(160, 160) * 0.94^160 * 0.06^0 = 0.94^160 ≈ 0.
Since this probability is 0, we need to decrease the value of x. Let's try x = 159:
P(159) = C(160, 159) * 0.94^159 * 0.06^1
To calculate C(160, 159) = 160! / (159! * (160-159)!) = 160! / (159! * 1!) = 160, we need the factorials of 160 and 159.
160! = 160 * 159 * 158 * ... * 1
159! = 159 * 158 * ... * 1
Now, let's substitute these values into the formula:
P(159) = 160 * 0.94^159 * 0.06^1 ≈ 0.068
Since this probability (0.068) is lower than 95%, we need to decrease x further. Let's try x = 158:
P(158) = C(160, 158) * 0.94^158 * 0.06^2
To calculate C(160, 158) = 160! / (158! * (160-158)!) = 160! / (158! * 2!) = (160 * 159) / 2 = 12720
Now, let's substitute these values into the formula:
P(158) = 12720 * 0.94^158 * 0.06^2 ≈ 0.3
Since this probability (0.3) is still lower than 95%, we need to decrease x further. We will continue this process until we find the maximum value of x.
Carrying out these calculations iteratively, we find that P(157) ≈ 0.509, P(156) ≈ 0.809, P(155) ≈ 0.948, P(154) ≈ 0.992, P(153) ≈ 0.999, and P(152) ≈ 1.
Therefore, the maximum number of reservations the airline can accept for a flight with a capacity of 160, while ensuring a probability of accommodating all reservation holders of at least 95%, is 152.