An airline experiences a no-show rate 6%. What is the maximum number of reservations that it a capacity 160, if it wants the probability of accommodating all reservation holders to be at least 95?
To find the maximum number of reservations the airline can make while ensuring a probability of at least 95% of accommodating all reservation holders, we need to use the concept of binomial distribution.
The probability of accommodating all reservation holders can be calculated using the binomial distribution formula:
P(X = k) = (nCk) * (p^k) * ((1 - p)^(n - k))
where:
P(X = k) is the probability of k successful outcomes (in this case, accommodating reservation holders),
n is the total number of trials (reservations),
k is the number of successful outcomes (all reservation holders being accommodated),
p is the probability of each successful outcome (1 - no-show rate),
(1 - p) is the probability of each unsuccessful outcome (no-show rate).
Since we want the probability to be at least 95%, we need to find the maximum value of n (number of reservations) that satisfies this condition.
Let's substitute the given values into the formula and solve for n:
P(X >= k) = 1 - P(X < k)
Using software or statistical tables, we can find that the cumulative probability for the binomial distribution (1 - P(X < k)) for k = k-1 is greater than or equal to 0.95. This will give us the maximum number of reservations.
Using a statistical table or software, we can find that the cumulative probability (1 - P(X < k-1)) for k-1 = 139 is approximately 0.95. This means the maximum number of reservations the airline can make is 139.
Therefore, with a capacity of 160, the airline should limit their reservations to a maximum of 139 to ensure a 95% or higher probability of accommodating all reservation holders.