Airlines sometimes overbook flights. Suppose that for a...

Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable $Y$ as the number of ticketed passengers who actually show up for the flight. The probability mass function of $Y$ appears in the accompanying table. (a) What is the probability that the flight will accommodate all ticketed passengers who show up? (b) What is the probability that not all ticketed passengers who show up can be accommodated? (c) If you are the first person on the standby list (which means you will be the first one to get on the plane if there are any seats available after all ticketed passengers have been accommodated), what is the probability that you will be able to take the flight? What is this probability if you are the third person on the standby list?

Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable $Y$ as the number of ticketed passengers who actually show up for the flight. The probability mass function of $Y$ appears in the accompanying table.
(a) What is the probability that the flight will accommodate all ticketed passengers who show up?
(b) What is the probability that not all ticketed passengers who show up can be accommodated?
(c) If you are the first person on the standby list (which means you will be the first one to get on the plane if there are any seats available after all ticketed passengers have been accommodated), what is the probability that you will be able to take the flight? What is this probability if you are the third person on the standby list?

Key Concepts

Discrete Random Variables

A discrete random variable is one that takes on at most a countable number of distinct values. In probability theory, these variables are used to model outcomes in scenarios where the results can be distinctly enumerated, and they are characterized by a probability mass function that assigns a probability to each value.

Probability Mass Function

The probability mass function (PMF) gives the probability that a discrete random variable is exactly equal to a particular value. It is fundamental in calculating probabilities for specific outcomes as well as cumulative probabilities by summing over ranges of outcomes.

Capacity Constraints and Overbooking

In many real-world applications such as transportation and reservation systems, a fixed capacity restricts the total number of individuals that can be accommodated. Overbooking involves selling more tickets than the available capacity, prompting the need to assess the probability that the number of arrivals does not exceed this limit. Understanding this concept is crucial in managing risks associated with excess demand.

Complement Rule

The complement rule is a basic principle in probability that states the probability of an event not occurring is equal to one minus the probability of the event occurring. This rule simplifies the process of determining the likelihood of the opposite or complementary event when direct computation is complex.

Conditional Probability in Sequential Ordering

Conditional probability deals with the likelihood of an event occurring given that another event has already occurred. In situations where order matters—such as determining which individuals on a waiting list may eventually secure a spot—the extension of conditional probability helps in evaluating the chances of success or failure relative to sequential positions.

Transcript

0:00 Today.

00:01 We're gonna be looking at the probability um having to do with airlines overbooking flights, meaning that they sell more seats than they're actually available.

00:08 Banking on the fact that some passengers will not show up.

00:11 And we know that that is definitely possible and we know the airlines do that because they are interested in making money.

00:19 Um so in this particular instance we have 50 seats on the plane and there are 55 tickets that are sold.

00:25 And then we have a probability um distribution given with the probability of a certain number of passengers actually showing up.

00:33 So part a.

00:34 We are looking what is the probability that the flight will accommodated accommodate all passing all ticket ticketed passengers who show up and that's going to be based off of our probability distribution.

00:46 The probability that are variables wa wise 45 plus the probability that y is 46 all the way up to the probability that wise 50 because of 46 passengers show up.

01:06 There's enough seats for everybody.

01:07 If 45 passengers show up, there's enough for everybody all the way up to 50.

01:13 And we're just gonna take those probabilities from our distribution and add those up just and that will give us our answer for um or a which is a pretty high percentage.

01:44 Okay, it's 83%.

01:46 So 83% of the time their lines find they don't have to give out any vouchers.

01:51 They don't have to try to make it right with anybody.

01:53 Everything's good.

01:54 Okay.

01:55 The passengers showed up.

01:56 There was a spot for him on the plane.

01:58 So then part b...