Problem 10 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Early in the morning, a group of \(m\) people decides to use the elevator in an otherwise deserted building of 21 floors. Each of these persons chooses his or her floor independently of the others, and-from our point of viewcompletely at random, so that each person selects a floor with probability \(1 / 21\). Let \(S_{m}\) be the number of times the elevator stops. In order to study \(S_{m}\), we introduce for \(i=1,2, \ldots, 21\) random variables \(R_{i}\), given by \(R_{i}= \begin{cases}1 & \text { if the elevator stops at the } i \text { th floor } \\ 0 & \text { if the elevator does not stop at the } i \text { th floor. }\end{cases}\) a. Each \(R_{i}\) has a \(\operatorname{Ber}(p)\) distribution. Show that \(p=1-\left(\frac{20}{21}\right)^{m}\). b. From the way we defined \(S_{m}\), it follows that $$ S_{m}=R_{1}+R_{2}+\cdots+R_{21} . $$ Can we conclude that \(S_{m}\) has a \(\operatorname{Bin}(21, p)\) distribution, with \(p\) as in part a? Why or why not? c. Clearly, if \(m=1\), one has that \(\mathrm{P}\left(S_{1}=1\right)=1\). Show that for \(m=2\) $$ \mathrm{P}\left(S_{2}=1\right)=\frac{1}{21}=1-\mathrm{P}\left(S_{2}=2\right) $$ and that \(S_{3}\) has the following distribution. $$ \begin{array}{cccc} a & 1 & 2 & 3 \\ \hline \mathrm{P}\left(S_{3}=a\right) & 1 / 441 & 60 / 441 & 380 / 441 \end{array} $$

Step-by-Step Solution

Answer

(a) Yes, (b) Yes, (c) Calculated as given.

Step 1: Determine Bernoulli distribution parameter

The random variable \( R_i \) is 1 if the elevator stops on the \( i \)-th floor, which happens if at least one person out of \( m \) chooses the \( i \)-th floor. The probability that a single person does not choose the \( i \)-th floor is \( \frac{20}{21} \). If \( m \) people choose floors independently, the probability that none of them chooses the \( i \)-th floor is \( \left(\frac{20}{21}\right)^m \). Therefore, the probability that the elevator stops at the \( i \)-th floor (at least one person chooses it) is \( p = 1 - \left(\frac{20}{21}\right)^m \).

Step 2: Sum of Bernoulli Variables

The sum \( S_m = R_1 + R_2 + \cdots + R_{21} \) represents the total count of distinct floors that the elevator stops at. Since each \( R_i \) is an independent Bernoulli random variable with success probability \( p \), \( S_m \) can be thought of as a binomial distribution with parameters \( n = 21 \) and \( p = 1 - \left(\frac{20}{21}\right)^m \).

Step 3: Validate if it is a Binomial Distribution

We can conclude \( S_m \) follows a Binomial distribution \( \text{Bin}(21, p) \) because it is a sum of 21 independent Bernoulli trials each with success probability \( p \). The criteria for a binomial distribution are met: fixed number of trials (21), each trial is independent, and each has the same probability \( p \).

Step 4: Calculate Probabilities for \( m=2 \)

With \( m=1 \), it is obvious that \( S_1 = 1 \) because one person chooses one floor. For \( m=2 \), only two cases: either both people choose the same floor, or they choose different floors. The probability both choose the same floor is \( \frac{1}{21} \). Therefore, the probability they choose different floors (\( S_2 = 2 \)) is complementary: \( 1 - \frac{1}{21} = \frac{20}{21} \).

Step 5: Calculate Distribution for \( S_3 \)

For \( m=3 \), let's derive the probabilities:- The event \( S_3 = 1 \) happens if all 3 choose the same floor, and there are 21 ways to choose a floor. Therefore, the probability is \( \frac{21}{21^3} = \frac{1}{441} \).- The event \( S_3 = 2 \) happens if 2 choose one floor and the 3rd person chooses a different floor. There are \( 21 \) choices for the pair's floor and \( 20 \) choices for the individual, totaling \( 21 \cdot 20 \cdot \binom{3}{2} = 1260 \) ways. The probability is \( \frac{1260}{9261} = \frac{60}{441} \).- For \( S_3 = 3 \), all 3 choose different floors. There are \( 21 \cdot 20 \cdot 19 \) ways, and the probability is \( \frac{7980}{9261} = \frac{380}{441} \).

Key Concepts

Bernoulli Distribution

The Bernoulli distribution is one of the simplest distributions in probability theory. It models a random process that has two possible outcomes: success or failure. In our example with the elevator, the random variable \( R_i \) represents the scenario where the elevator either stops (success, \( R_i = 1 \)) on the \( i \)-th floor or does not stop (failure, \( R_i = 0 \)).

To determine whether the elevator stops on a given floor, we calculate the probability that at least one person chooses it. If the probability each person selects a particular floor is independent and equal to \( \frac{1}{21} \), then the probability that no one selects that floor is \( \left( \frac{20}{21} \right)^m \). Conversely, the probability that at least one person chooses that floor and the elevator stops there is a success, given by \( p = 1 - \left( \frac{20}{21} \right)^m \).

This structure fits perfectly into the framework of the Bernoulli distribution, where each floor check is an independent trial with two possible outcomes, making it a cornerstone for understanding more complex distributions like the binomial distribution.

Binomial Distribution

The binomial distribution is essentially a series of repeated Bernoulli trials. Each trial is independent, and there are a fixed number of trials. In the elevator scenario, our trials happen each time we check whether the elevator stops on a particular floor out of the 21 possible. The total number of successful stops is the sum \( S_m = R_1 + R_2 + \cdots + R_{21} \).

This sum characterizes a binomial distribution, \( \operatorname{Bin}(n, p) \), where \( n \) is the number of trials (21 floors) and \( p \) is the probability of success per trial (probability that the elevator stops at a floor). Since each \( R_i \) represents an independent Bernoulli trial, \( S_m \) is binomially distributed with parameters \( n = 21 \) and \( p = 1 - \left( \frac{20}{21} \right)^m \).

This distribution allows us to compute the probability of different numbers of stops that the elevator can make, giving insight into stopping patterns across occupants choosing floors independently.

Random Variables

Random variables are fundamental in probability theory and statistics, representing numeric outcomes of different random events. In this problem, the random variable \( R_i \) indicates whether the elevator stops at the \( i \)-th floor. When you consider all 21 floors, these form a collection of 21 random variables.

Each \( R_i \) can independently take the values 0 or 1 (the Bernoulli outcome), demonstrating how random variables can simplify the representation and analysis of probabilistic situations. When summed together, as in \( S_m = R_1 + R_2 + \cdots + R_{21} \), these random variables reflect the total number of stops made by the elevator. The behavior of \( S_m \), therefore, is governed by the characteristics and properties of the random variables \( R_i \).

Understanding random variables is crucial, as they allow us to build complex models and make informed conclusions in scenarios of uncertainty based on probabilities.

Independence in Probability

Independence is a critical concept in probability theory, particularly when dealing with probabilities of multiple events. When we say that the random selections of floors by different people are independent, it means that the choice made by one person does not affect the choice made by another.

In terms of the elevator problem, independence is crucial for validly applying the binomial distribution. Each \( R_i \) being independent ensures that the probability of stopping at one floor does not interfere with the probability of stopping at any other floor.

The independence assumption simplifies our analysis significantly. It permits us to use the product rule to determine the overall probability of the elevator stopping on any given floor, as no floor\’s probability depends on another. Therefore, independence not only simplifies calculations but also helps in developing probabilistic models that represent real-world scenarios accurately.