Question

There are $n$ people eligible to vote in a certain election. Voting requires registration. Decisions are made independently. Each of the $n$ people will register with probability $p_{1}$. Given that a person registers, they will vote with probability $p_{2}$. Given that a person votes, they will vote for Kodos (who is one of the candidates) with probability $p_{3}$. What is the distribution of the number of votes for Kodos (give the PMF, fully simplified, or the name of the distribution, including its parameters)?

Name: Problem 23, Chapter 3: Introduction to Probability
Uploaded: 2023-01-23T23:54:35-08:00
Duration: 2 min 3 s
Channel: James Kiss

   There are $n$ people eligible to vote in a certain election. Voting requires registration. Decisions are made independently. Each of the $n$ people will register with probability $p_{1}$. Given that a person registers, they will vote with probability $p_{2}$. Given that a person votes, they will vote for Kodos (who is one of the candidates) with probability $p_{3}$. What is the distribution of the number of votes for Kodos (give the PMF, fully simplified, or the name of the distribution, including its parameters)?

Introduction to Probability

Joseph K.… 2nd Edition

Chapter 3, Problem 23

Instant Answer

Solved by Expert James Kiss

01/23/2023

Step 1

Since the decisions are made independently, we can multiply the probabilities at each step. So, the probability that a person will vote for Kodos is $p_1 \cdot p_2 \cdot p_3$. Now, let's consider the distribution of the number of votes for Kodos. Since there are Show more…

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There are $n$ people eligible to vote in a certain election. Voting requires registration. Decisions are made independently. Each of the $n$ people will register with probability $p_{1}$. Given that a person registers, they will vote with probability $p_{2}$. Given that a person votes, they will vote for Kodos (who is one of the candidates) with probability $p_{3}$. What is the distribution of the number of votes for Kodos (give the PMF, fully simplified, or the name of the distribution, including its parameters)?

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Key Concepts

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. Here, each person’s decision to vote for a candidate is seen as a success with overall probability p1 × p2 × p3, so the total number of votes follows a binomial distribution with parameters n and p1*p2*p3.

Bernoulli Trials

A Bernoulli trial is an experiment with exactly two outcomes: success or failure. In this context, each person goes through a series of independent decisions where the final outcome (voting for a given candidate) can be considered a success if all conditions are met. The sequence of independent Bernoulli trials over n individuals forms the basis of the binomial distribution.

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