91Ó°ÊÓ

In some states, such as Iowa and Nevada, the presidential primaries are decided by caucuses rather than a primary election. The caucuses determine winners at the precinct level, and turnout is often low. As a result, it is not uncommon in a close race to have some caucuses end in a tie. The article "A Nevada Tie to Be Decided by Cards" (The Wall Street Journal, February 20,2016 ) reported that in Nevada a tie is decided by having each side draw a card, with the high card winning. In Iowa, a tie is decided by a coin toss. In 2016 , in the primary race between Hillary Clinton and Bernie Sanders, some Democratic caucuses were in fact decided by coin tosses. a. Suppose two caucuses resulted in a tie between Bernie Sanders and Hillary Clinton. What is the probability that both would be decided in favor of Hillary Clinton? b. Suppose two caucuses resulted in a tie between Bernie Sanders and Hillary Clinton. What is the probability that both would be decided in favor of Bernie Sanders? c. Suppose two caucuses resulted in a tie between Bernie Sanders and Hillary Clinton. What is the probability that both would be decided in favor of the same candidate? d. Suppose that three caucuses resulted in a tie between Bernie Sanders and Hillary Clinton. What is the probability that all three caucuses would be decided in favor of the same candidate?

Step by step solution

01

a. Probability of both caucuses decided in favor of Hillary Clinton

In this problem, we need to find the probability that both caucuses would be decided in favor of Hillary Clinton. Since each caucus has two possible outcomes (win or lose for Hillary Clinton), the probability of her winning a single caucus is 1/2, or 0.5. Since we want to find the probability that she wins both caucuses, we can multiply the probability of each of the two events: \(P(Hillary \ wins \ both \ caucuses)\) \(= P(Hillary \ wins \ 1st \ caucus) \times P(Hillary \ wins \ 2nd \ caucus)\) \(= 0.5 \times 0.5\) \(= 0.25\)

02

b. Probability of both caucuses decided in favor of Bernie Sanders

Similarly, we can find the probability that both caucuses would be decided in favor of Bernie Sanders. Since each caucus has two possible outcomes (win or lose for Bernie Sanders), the probability of him winning a single caucus is 1/2, or 0.5. Since we want to find the probability that he wins both caucuses, we can multiply the probability of each of the two events: \(P(Bernie \ wins \ both \ caucuses)\) \(= P(Bernie \ wins \ 1st \ caucus) \times P(Bernie \ wins \ 2nd \ caucus)\) \(= 0.5 \times 0.5\) \(= 0.25\)

03

c. Probability of both caucuses decided in favor of the same candidate

In this problem, we need to find the probability that both caucuses would be decided in favor of the same candidate. To do this, we can add the probabilities of both candidates winning both caucuses: \(P(Same \ candidate\ wins \ both)\) \(= P(Hillary \ wins \ both) + P(Bernie \ wins \ both)\) \(= 0.25 + 0.25\) \(= 0.5\)

04

d. Probability of all three caucuses decided in favor of the same candidate

Now, we need to find the probability that all three caucuses would be decided in favor of the same candidate. First, we'll calculate the probabilities that each candidate wins all three caucuses: \(P(Hillary \ wins\ all\ 3)\) \(= P(Hillary \ wins\ 1st) \times P(Hillary \ wins\ 2nd) \times P(Hillary \ wins\ 3rd)\) \(= 0.5 \times 0.5 \times 0.5\) \(= 0.125\) \(P(Bernie \ wins\ all\ 3)\) \(= P(Bernie \ wins\ 1st) \times P(Bernie \ wins\ 2nd) \times P(Bernie \ wins\ 3rd)\) \(= 0.5 \times 0.5 \times 0.5\) \(= 0.125\) Next, we can add the probabilities of both candidates winning all three caucuses: \(P(Same \ candidate\ wins\ all\ 3)\) \(= P(Hillary \ wins\ all\ 3) + P(Bernie \ wins\ all\ 3)\) \(= 0.125 + 0.125\) \(= 0.25\)

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Key Concepts

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

Probability Theory

Probability theory is the branch of mathematics that deals with calculating the likelihood of various outcomes. It's essential for understanding events where the outcome is uncertain, and it forms the foundation for statistics and many methods used in data analysis. The core idea behind probability is quantifying how likely an event is to occur, usually expressed as a number between 0 and 1.

In political science, and specifically in the context of political caucuses, probability theory helps to compare the chances of different scenarios, like which candidate will win a caucus. For example, if a caucus ends in a tie and the outcome is decided by a coin toss, the probability of either candidate winning is equal - demonstrating one of the simplest probability scenarios with two equally likely outcomes.

Political Caucuses

A political caucus is a local gathering where members of a political party meet to select delegates who will vote for a presidential nominee at a later convention. Unlike primaries, caucuses are often characterized by active and public participation, where supporters discuss and persuade each other before the final vote. As the exercise from the textbook shows, certain states like Iowa and Nevada use this system during presidential nominations.

Understanding the unique aspects of caucuses, such as their low turnout and the potential for ties leading to coin tosses or card draws, adds complexity to predicting outcomes. This necessity to resolve ties randomly is a prime example of probability in action within the context of political science.

Statistical Outcomes

Statistical outcomes are the results you get from applying statistical methods to a set of data. These outcomes are often probabilities or predictions about what will happen based on data trends. In political caucuses, if there's enough historical data, statisticians can predict turnout, the likelihood of ties, and even estimate the distribution of votes.

To make these predictions, we often use probability distribution functions. The tiebreaker methods (coin tosses or card draws) presented in the textbook exercise show the need to understand the basic statistical outcome of equally likely events. It helps make sense not only of the chances of one candidate winning, but also the implications of repeated independent events. For example, calculating the chance of the same candidate winning consistently over several caucuses.