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 that an airline can accommodate, given a certain capacity and a desired probability, we can use the concept of binomial distribution.
The binomial distribution is a discrete probability distribution that measures the number of successful outcomes in a fixed number of independent Bernoulli trials (success or failure). In this case, each reservation can be considered a Bernoulli trial, where success is defined as a reservation holder showing up for their flight and failure is defined as a reservation holder being a no-show.
To solve this problem, we can set up the binomial distribution as follows:
n = Number of reservations
p = Probability of success (reservation holder shows up) = 1 - Probability of failure (reservation holder is a no-show) = 1 - 0.06 = 0.94
k = Number of successful outcomes (all reservation holders show up)
P(X=k) = Probability of k successful outcomes in n trials
We want to find the maximum number of reservations (n) for which the probability of accommodating all reservation holders (P(X=n)) is at least 95%. In other words, we want to find the smallest n for which P(X≥n) ≥ 0.95.
To calculate this, we can use cumulative probability (P(X≥n)) from a binomial distribution table or a statistical software. However, I will use a simple trial and error approach to approximate the maximum number of reservations.
Starting from n = 1, we can calculate the cumulative probability and check if it is greater than or equal to 0.95. If it is, we have found the maximum number of reservations. If not, we increment n by 1 and repeat the process until we find the maximum value.
Let's go step by step:
Step 1: Initialize n = 1.
Step 2: Calculate the cumulative probability P(X≥n) using the binomial distribution formula or a calculator.
- For n=1, P(X≥1) = 1 - P(X=0) = 1 - (0.94^1 * 0.06^0) = 1 - 0.06 = 0.94.
Step 3: Check if P(X≥n) ≥ 0.95.
- In this case, 0.94 < 0.95, so we move to the next step.
Step 4: Increment n by 1.
- n = 1 + 1 = 2.
Step 5: Calculate the cumulative probability P(X≥n).
- For n=2, P(X≥2) = 1 - P(X=0) - P(X=1) = 1 - (0.94^2 * 0.06^0) - (0.94^1 * 0.06^1) = 1 - 0.8836 - 0.0564 = 0.06.
Step 6: Check if P(X≥n) ≥ 0.95.
- In this case, 0.94 < 0.9552, so we move to the next step.
Step 7: Repeat steps 4-6 until P(X≥n) ≥ 0.95.
- Continuing with n = 3, P(X≥3) = 0.06324
- Continuing with n = 4, P(X≥4) = 0.06775
- Continuing with n = 5, P(X≥5) = 0.07260
- Continuing with n = 6, P(X≥6) = 0.07781
- Continuing with n = 7, P(X≥7) = 0.08341
- Continuing with n = 8, P(X≥8) = 0.08942
- Continuing with n = 9, P(X≥9) = 0.09586
At n = 9, P(X≥9) = 0.09586, which is greater than or equal to 0.95. Therefore, the maximum number of reservations that the airline can accommodate, given a capacity of 160 and a desired probability of at least 95%, is 9.