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Chapter 5: Problem 78

Urban voters The voters in a large city are \(40 \%\) white, \(40 \%\) black, and \(20 \%\) Hispanic. (Hispanics may be of any race in official statistics, but here we are speaking of political blocks. A mayoral candidate anticipates attracting \(30 \%\) of the white vote, \(90 \%\) of the black vote, and \(50 \%\) of the Hispanic vote. Suppose we select a voter at random. (a) Draw a tree diagram to represent this situation. (b) Find the probability that this voter votes for the mayoral candidate. Show your work. (c) Given that the chosen voter plans to vote for the candidate, find the probability that the voter is black. Show your work.

Short Answer

Expert verified
(b) Probability of voting for the candidate is 0.58. (c) Probability that voter is black given they vote for the candidate is approximately 0.6207.

Step by step solution

01

Understanding the Voter Distribution

The problem provides us with the distribution of voters by political block: 40% are white, 40% are black, and 20% are Hispanic. These probabilities will help structure our tree diagram and subsequent probability calculations.
02

Construct the Tree Diagram

The tree diagram needs to represent both the total population by political block (first level) and the proportion who vote for the candidate within each block (second level). Start by drawing the branches for the white (40%), black (40%), and Hispanic (20%) voter groups. Then add branches to show those voting for the candidate: 0.3 for white, 0.9 for black, and 0.5 for Hispanic.
03

Calculate Probability of Voting for the Candidate

To find this probability, use the tree diagram. For each voter group, multiply the proportion of the population by the proportion voting for the candidate, then sum these products:\[ P(V_{W}) = 0.4 \times 0.3 = 0.12 \]\[ P(V_{B}) = 0.4 \times 0.9 = 0.36 \]\[ P(V_{H}) = 0.2 \times 0.5 = 0.10 \] Adding these probabilities gives the total probability of a randomly selected voter voting for the candidate:\[ P(V) = 0.12 + 0.36 + 0.10 = 0.58 \]
04

Calculate Conditional Probability of Being Black Given Voting for Candidate

To find this probability, use the formula for conditional probability:\[ P(B | V) = \frac{P(B \cap V)}{P(V)} \] We already know from Step 3 that \( P(V) = 0.58 \). The probability that a voter is both black and votes for the candidate is \( P(B \cap V) = 0.36 \). Therefore,\[ P(B | V) = \frac{0.36}{0.58} \approx 0.6207 \]

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Key Concepts

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

Tree Diagram
A Tree Diagram is a visual representation that helps in organizing possible outcomes and their probabilities. It resembles a tree with branches representing different outcomes. In our case, the tree diagram organizes the voters based on their political blocks and their likelihood of voting for a particular mayoral candidate.

Here's how you can create a tree diagram for the urban voters:
  • Start with a single point. From this point, draw three branches representing each political block: white, black, and Hispanic.
  • Label each branch with the probability of a voter belonging to that group: 0.4 for white, 0.4 for black, and 0.2 for Hispanic.
  • From each of these branches, draw sub-branches to show the probability of a voter from each group voting for the candidate. Indicate these probabilities as 0.3, 0.9, and 0.5 respectively.
The tree diagram makes it easy to visually calculate outcomes by following path probabilities. It illustrates not only the probabilities at each stage but also how final outcomes are achieved through combinations of initial choices.
Conditional Probability
Conditional Probability is the probability of an event occurring given that another event has already occurred. In this context, we're interested in finding the probability that a voter is black, given they are voting for the candidate.

The formula for conditional probability is: \[ P(A | B) = \frac{P(A \cap B)}{P(B)} \] Here, \( P(A | B) \) represents the probability of event A occurring given B. For our exercise:
  • Let A be the event that a voter is black: this is \( P(B) = 0.4 \).
  • Let B be the event that the voter votes for the candidate: this is \( P(V) = 0.58 \).
  • The joint probability \( P(B \cap V) = 0.36 \), comes from the probability a voter is both black and votes for the candidate.
Substituting these into the formula gives \( P(B | V) = \frac{0.36}{0.58} \approx 0.6207 \).

This means if a voter is known to vote for the candidate, there is a 62% chance that this voter is black.
Voter Distribution
Understanding voter distribution is crucial in probability exercises as it sets the groundwork for constructing probabilities correctly.

Voter distribution refers to how various groups in a population are spread out or represented. For the problem at hand:
  • Forty percent of the voters are white, making up a significant part of the voting demographic.
  • Similarly, forty percent are black, which means they equally weigh on the election outcomes as their voting percentage matches.
  • The remaining twenty percent are Hispanic, representing a smaller yet significant voting bloc.
In practical terms, voter distribution helps in understanding how different segments of the population might impact election results. It is the first critical step in configuring tree diagrams and performing calculations related to conditional probabilities, as seen in the previous sections.

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Most popular questions from this chapter

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