An airport limousine can accommodate up to four passengers...

An airport limousine can accommodate up to four passengers on any one trip. The company will accept a maximum of six reservations for a trip, and a passenger must have a reservation. From previous records, 20$\%$ of all those making reservations do not appear for the trip. In the following questions, assume independence, but explain why there could be dependence. (a) If six reservations are made, what is the probability that at least one individual with a reservation cannot be accommodated on the trip? (b) If six reservations are made, what is the expected number of available places when the limousine departs? (c) Suppose the probability distribution of the number of reservations made is given in the accompanying table. Let $X$ denote the number of passengers on a randomly selected trip. Obtain the probability mass function of $X .$

An airport limousine can accommodate up to four passengers on any one trip. The company will accept a maximum of six reservations for a trip, and a passenger must have a reservation. From previous records, 20$\%$ of all those making reservations do not appear for the trip. In the following questions, assume independence, but explain why there could be dependence.
(a) If six reservations are made, what is the probability that at least one individual with a reservation cannot be accommodated on the trip?
(b) If six reservations are made, what is the expected number of available places when the limousine departs?
(c) Suppose the probability distribution of the number of reservations made is given in the accompanying table.
Let $X$ denote the number of passengers on a randomly selected trip. Obtain the probability mass function of $X .$

Key Concepts

Probability Mass Function (PMF) and Mixture Distributions

The probability mass function (PMF) describes the probabilities of all possible outcomes for a discrete random variable. When the total number of reservations is itself a random variable, combining this with the binomial outcomes of whether a reserved passenger shows up creates a mixture distribution. This compound PMF is obtained using the law of total probability, integrating over all possible numbers of reservations.

Independence and Dependence in Events

Assuming independence means that the occurrence of one event (such as a reservation showing up) does not affect another. However, in reality, factors such as weather or transportation issues can create dependence between events, meaning that if one passenger misses their reservation, others might be similarly affected. Recognizing potential dependencies is important for accurate modeling of actual outcomes.

Overbooking and No-Show Behavior

Overbooking is the practice of accepting more reservations than available capacity based on historical data of no-shows. The notion of no-show behavior, where each reservation has a fixed probability of not resulting in attendance, helps businesses optimize occupancy while managing the risk of overselling, provided that the statistics are reliable and the events are approximately independent.

Binomial Distribution

The binomial distribution is used to model the number of successes in a fixed number of independent trials when each trial has the same probability of success. In the context of the problem, each reservation is considered an independent trial where a passenger either shows up (success) or does not (failure), with a given probability.

Capacity Constraints and Truncation

Capacity constraints refer to the upper limit on the number of individuals that can be accommodated. When modeling real-world scenarios, any excess beyond the capacity is effectively truncated, meaning that even if more individuals show up, only up to the maximum capacity can be served. This idea becomes crucial when calculating the number of available spaces or denials of service.

Transcript

00:01 I will now give my explanation to part a.

00:05 Recore that the probability distribution of the number of successes obtained is called binomial probability distribution.

00:14 Here's the binomial distribution formula, where n is the number of trials.

00:19 For our problem, six reservations are made, so n equals 6.

00:27 X is the number of successes desired.

00:30 In our case, will be the number of people with reservations.

00:33 That show up.

00:36 P is the probability of getting a success in one trial.

00:42 20 % of those making reservations do not appear for the trip, so the probability that they do appear is 1 minus 0 .2 equals 0 .8.

00:57 And finally, q is the probability of getting a failure in one trial.

01:03 Which is 0 .2.

01:07 A problem states that an airport limousine can accommodate up to four passengers on any one trip.

01:15 So let us visualize this.

01:18 The limousine has four seats and can accommodate four people.

01:38 If at least one person show up, then he or she cannot be accommodated and we will have at least 5 people in total.

01:54 So, having at least one person with a reservation that cannot be accommodated means at least 5 passengers will show up and that probability will be p of x larger or equal to 5 which equals p of x equal 5 plus p of x equals 6.

02:23 We just plug in the number into our binomial distribution formula and we'll get this the following.

03:18 And this is our final answer.

03:32 Let us move to part b.

03:36 Here's the formula for the expected value of random variable y.

03:42 For our problem, y is the number of empty seats and and p y equals y is the probability that a trip has y empty seats.

03:54 Now let us construct the following table.

04:03 Where the first column is y, the second column is the number of pupil that show up...

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