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Chapter 13: Problem 43

Suppose that in your city \(37 \%\) of the voters are registered as Democrats, \(29 \%\) as Republicans, and \(11 \%\) as members of other parties (Liberal, Right to Life, Green, etc.). Voters not aligned with any official party are termed "Independent." You are conducting a poll by calling registered voters at random. In your first three calls, what is the probability you talk to a) all Republicans? b) no Democrats? c) at least one Independent?

Short Answer

Expert verified
a) 0.024, b) 0.250, c) 0.543

Step by step solution

01

Define the Problem

We need to calculate the probabilities for three different events based on the given voter registration percentages. We have three probability values: Democrats (37%), Republicans (29%), and other parties or Independents making up the rest. We need to derive each event scenario using these probabilities.
02

Calculate Probability of All Republicans

To find the probability that all three called voters are Republicans, we use the probability of calling one Republican voter. Let this probability be \( P(R) = 0.29 \). The probability of calling three Republicans in a row is \( P(R)^3 \). Therefore, \[ P( ext{All Republicans}) = 0.29^3 = 0.024389 \]
03

Calculate Probability of No Democrats

For no Democrats, we first need the probability of not calling a Democrat. This is equivalent to calling either a Republican or any other voter. The combined probability of not getting a Democrat is:\[ P( ext{Not Democrat}) = 1 - 0.37 = 0.63 \]The probability of all three not being Democrats is:\[ P( ext{No Democrats}) = 0.63^3 = 0.250047 \]
04

Calculate Probability of At Least One Independent

First, we calculate the probability of not calling an Independent in three attempts. Independents make up the remaining portion after accounting for Democrats and Republicans, so \[ P( ext{Independent}) = 1 - (0.37 + 0.29 + 0.11) = 0.23 \]Probability of not calling an Independent:\[ P( ext{Not Independent}) = 1 - 0.23 = 0.77 \]The probability none of the three called is Independent is:\[ 0.77^3 = 0.456533 \]Finally, the probability of at least one Independent is:\[ P( ext{At least one Independent}) = 1 - 0.456533 = 0.543467 \]

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Key Concepts

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

Understanding Voter Registration
Voter registration is the process by which eligible citizens enroll themselves to vote in elections. It's crucial for maintaining an organized and democratic process. In our scenario, we know the distribution of registered voters: 37% are Democrats, 29% are Republicans, and 11% belong to other minor parties. The remaining percentage of voters, those who aren't affiliated with these official parties, are known as Independents. This distribution is essential for computing the probability of encountering different types of voters in a survey or poll, as illustrated in our exercise.
Fundamentals of Event Probability
When calculating probabilities, we're determining the likelihood of certain events happening. In terms of voter registration, each phone call to a voter is considered an independent event. For instance, to find the probability that all called voters are Republicans in three calls, we would multiply the probability of speaking to one Republican by itself three times, as each call is independent of others. Generally, if an event can happen in a certain way, the probability is calculated by multiplying the individual probabilities for those combinations.
Who are Independent Voters?
Independent voters are individuals who do not align themselves with a specific party like Democrats or Republicans. They can be pivotal in elections due to their ability to support various candidates based on issues rather than party lines. In our scenario, we determined the percentage of Independent voters by subtracting the known percentages of Democrats, Republicans, and other parties from 100%. Understanding the role of Independents in probability helps anticipate polling outcomes and adjust strategies accordingly.
Democrats and Republicans in Probability
In our scenario, Democrats and Republicans are the two major registered parties. Knowing their percentages helps calculate the probability of particular voter interactions when conducting polls. If you call registered voters randomly, knowing these percentages aids in predicting outcomes such as the probability of bypassing Democrats or reaching only Republicans. For instance, the probability of engaging only Republicans in three calls leverages the independent probability of reaching a Republican with each call. Such calculations form a foundational part of understanding the makeup of voters for targeted surveys or political strategies.

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Most popular questions from this chapter

On February \(11,2009,\) the AP news wire released the following story: (LAS VEGAS, Nev.) - A man in town to watch the NCAA basketball tournament hit a \(\$ 38.7\) million jackpot on Friday, the biggest slot machine payout ever. The 25 -year-old software engineer from Los Angeles, whose name was not released at his request, won after putting three \(\$ 1\) coins in a machine at the Excalibur hotel-casino, said Rick Sorensen, a spokesman for slot machine maker International Game Technology. a) How can the Excalibur afford to give away millions of dollars on a \(\$ 3\) bet? b) Why was the maker willing to make a statement? Wouldn't most businesses want to keep such a huge loss quiet?

You bought a new set of four tires from a manufacturer who just announced a recall because \(2 \%\) of those tires are defective. What is the probability that at least one of yours is defective?

Census reports for a city indicate that \(62 \%\) of residents classify themselves as Christian, \(12 \%\) as Jewish, and \(16 \%\) as members of other religions (Muslims, Buddhists, etc.). The remaining residents classify themselves as nonreligious. A polling organization seeking information about public opinions wants to be sure to talk with people holding a variety of religious views, and makes random phone calls. Among the first four people they call, what is the probability they reach a) all Christians? b) no Jews? c) at least one person who is nonreligious?

The Masterfoods company says that before the introduction of purple, yellow candies made up \(20 \%\) of their plain M\&M's, red another \(20 \%,\) and orange, blue, and green each made up \(10 \% .\) The rest were brown. a) If you pick an \(\mathrm{M} \& \mathrm{M}\) at random, what was the probability that 1) it is brown? 2) it is yellow or orange? 3) it is not green? 4) it is striped? b) If you pick three M\&M's in a row, what is the probability that 1) they are all brown? 2) the third one is the first one that's red? 3) none are yellow? 4) at least one is green?

You shuffle a deck of cards and then start turning them over one at a time. The first one is red. So is the second. And the third. In fact, you are surprised to get 10 red cards in a row. You start thinking, "The next one is due to be black!" a) Are you correct in thinking that there's a higher probability that the next card will be black than red? Explain. b) Is this an example of the Law of Large Numbers? Explain.
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