91Ó°ÊÓ

Airlines sometimes overbook flights. Suppose that for a plane with 100 seats, an airline takes 110 reservations. Define the random variable \(x\) as \(x=\) the number of people who actually show up for a sold-out flight on this plane From past experience, the probability distribution of \(x\) is given in the following table: $$ \begin{array}{|cc|} \hline \boldsymbol{x} & \boldsymbol{p}(\boldsymbol{x}) \\ \hline 95 & 0.05 \\ 96 & 0.10 \\ 97 & 0.12 \\ 98 & 0.14 \\ 99 & 0.24 \\ 100 & 0.17 \\ 101 & 0.06 \\ 102 & 0.04 \\ 103 & 0.03 \\ 104 & 0.02 \\ 105 & 0.01 \\ 106 & 0.005 \\ 107 & 0.005 \\ 108 & 0.005 \\ 109 & 0.0037 \\ 110 & 0.0013 \\ \hline \end{array} $$ a. What is the probability that the airline can accommodate everyone who shows up for the flight? b. What is the probability that not all passengers can be accommodated? c. If you are trying to get a seat on such a flight and you are number 1 on the standby list, what is the probability that you will be able to take the flight? What if you are number 3 ?

Step by step solution

01

Summarize Probabilities for x ≤ 100

Calculate the sum of probabilities for x = 95, 96, 97, 98, 99, and 100 from the given probability distribution table: \( P(x \leq 100) = P(95) + P(96) + P(97) + P(98) + P(99) + P(100) \) \( P(x \leq 100) = 0.05 + 0.10 + 0.12 + 0.14 + 0.24 + 0.17 \)

02

Calculate Probability

Now add the values to find the probability: \( P(x \leq 100) = 0.82 \) This means that there is a 82% chance that the airline can accommodate everyone who shows up for the flight. #b. Probability that not all passengers can be accommodated# This is the complementary event of part a, so we can calculate this probability as:

03

Calculate Complementary Probability

\( P(x > 100) = 1 - P(x \leq 100) \) \( P(x > 100) = 1 - 0.82 \) \( P(x > 100) = 0.18 \) There is an 18% chance that not all passengers can be accommodated. #c. Probability that a passenger on the standby list can take the flight# Let's first find the probability for a passenger who is number 1 on the standby list.

04

Summarize Probabilities for Cases When Standby Passenger 1 Gets a Seat

If standby passenger number 1 can take the flight, that means x ≤ 101. Calculate the sum of probabilities for x ∈ {95, 96, 97, 98, 99, 100, 101}: \( P(x \leq 101) = P(95) + P(96) + P(97) + P(98) + P(99) + P(100) + P(101) \) \( P(x \leq 101) = 0.82 + 0.06 \)

05

Calculate Probability

Now add the values to find the probability: \( P(x \leq 101) = 0.88 \) Therefore, the passenger number 1 on the standby list has an 88% chance of being able to take the flight. Now let's find the probability for a passenger who is number 3 on the standby list.

06

Summarize Probabilities for Cases When Standby Passenger 3 Gets a Seat

If standby passenger number 3 can take the flight, that means x ≤ 99. Calculate the sum of probabilities for x ∈ {95, 96, 97, 98, 99}: \( P(x \leq 99) = P(95) + P(96) + P(97) + P(98) + P(99)\) \( P(x \leq 99) = 0.05 + 0.10 + 0.12 + 0.14 + 0.24 \)

07

Calculate Probability

Now add the values to find the probability: \( P(x \leq 99) = 0.65 \) Therefore, the passenger number 3 on the standby list has a 65% chance of being able to take the flight.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Random Variables

In probability and statistics, a random variable is a variable that represents a numerical outcome of a random phenomenon. For instance, when we're considering the number of people who actually show up for a flight, denoted as variable 'x', we're dealing with a random variable because it quantifies an uncertain event. Random variables can be discreet, taking on separate values like the number of customers entering a shop each day, or continuous, like the amount of time until the next phone call at a call center.

When we look at the provided overbooking example, the random variable 'x' captures the number of passengers showing up for the 100-seat flight. This random variable could theoretically take on values from 0 (no one shows up) to 110 (everyone shows up), but practical experience might show that only a certain range of outcomes, such as 95 to 110, is realistic. The power of using a random variable lies in its ability to help us predict outcomes and make data-driven decisions in uncertain situations like flight overbooking.

Overbooking Flights Statistics

Overbooking is a common practice for airlines, driven by the motivation to maximize revenue since not all passengers always show up for their flights. In our example, the airline overbooks by 10% expecting that not all 110 passengers will come. This is informed by overbooking flights statistics, which are historical data showing the probability of different numbers of passengers showing up.

These statistics, typically represented as a probability distribution like in the table provided, empower airlines to calculate risks and make policy decisions. For example, knowing there's an 82% chance of accommodating all passengers allows the airline to weigh the financial benefit of selling extra tickets against the cost of potential compensation for bumped passengers. Overbooking decisions can have sizeable economic implications depending on how these probabilities play out in reality.

Complementary Probability

The concept of complementary probability is a foundational idea in probability theory, referring to the combined probability of all outcomes that are not part of the event of interest. In essence, it answers the question: 'If this event does not happen, what is the likelihood that anything else happens?'. Complementary probabilities always sum to 1 – if an event has a 70% chance of occurring, the complementary event has a 30% chance.

In our flight example, we determined an 82% chance that the airline can accommodate all passengers. Using complementary probability, the chance that they cannot accommodate everyone is 18%. This also means that a standby passenger would have an 18% or lower chance of getting a seat, depending on their position on the standby list. Understanding complementary probability helps individuals analyze scenarios with binary outcomes and assists in making informed decisions under uncertainty.