Questions: Airlines sometimes overbook flights. Suppose that for a plane with 100 seats, an airline takes 110 reservations. Define the variable x as the number of people who actually show up for a sold-out flight. From past experience, the population distribution of x is given in the following table: x P(x) 96 0.15 98 0.25 100 0.41 102 0.12 104 0.04 106 0.015 108 0.01 110 0.005 What is the probability that all the passengers who showed up can be accommodated?

Airlines sometimes overbook flights. Suppose that for a plane with 100 seats, an airline takes 110 reservations. Define the variable x as the number of people who actually show up for a sold-out flight. From past experience, the population distribution of x is given in the following table:

x P(x)
96 0.15
98 0.25
100 0.41
102 0.12
104 0.04
106 0.015
108 0.01
110 0.005

What is the probability that all the passengers who showed up can be accommodated?

Transcript text: Airlines sometimes overbook flights. Suppose that for a plane with 100 seats, an airline takes 110 reservations. Define the variable $x$ as the number of people who actually show up for a sold-out flight. From past experience, the population distribution of $x$ is given in the following table: \begin{tabular}{|l|l|} \hline$x$ & $P(x)$ \\ \hline 96 & 0.15 \\ \hline 98 & 0.25 \\ \hline 100 & 0.41 \\ \hline 102 & 0.12 \\ \hline 104 & 0.04 \\ \hline 106 & 0.015 \\ \hline 108 & 0.01 \\ \hline 110 & 0.005 \\ \hline \end{tabular} What is the probability that all the passengers who showed up can be accommodated?

Solution

Solution Steps

To find the probability that all passengers who show up can be accommodated, we need to calculate the probability that the number of people who show up is less than or equal to the number of seats available on the plane (100 seats). This involves summing the probabilities of all scenarios where the number of people who show up is 100 or fewer.

Step 1: Define the Probability Distribution

The probability distribution of the number of passengers $ x $ who show up is given as follows:

\[ \begin{array}{|c|c|} \hline x & P(x) \\ \hline 96 & 0.15 \\ 98 & 0.25 \\ 100 & 0.41 \\ 102 & 0.12 \\ 104 & 0.04 \\ 106 & 0.015 \\ 108 & 0.01 \\ 110 & 0.005 \\ \hline \end{array} \]

Step 2: Identify the Condition for Accommodation

To determine the probability that all passengers who show up can be accommodated, we need to find the probability that $ x \leq 100 $.

Step 3: Calculate the Probability

We sum the probabilities for all values of $ x $ that are less than or equal to 100:

\[ P(x \leq 100) = P(96) + P(98) + P(100) = 0.15 + 0.25 + 0.41 = 0.81 \]