91Ó°ÊÓ

A Pew Research survey asked 2,373 randomly sampled registered voters their political affiliation (Republican, Democrat, or Independent) and whether or not they identify as swing voters. \(35 \%\) of respondents identified as Independent, \(23 \%\) identified as swing voters, and \(11 \%\) identified as both. \(^{21}\) (a) Are being Independent and being a swing voter disjoint, i.e. mutually exclusive? (b) Draw a Venn diagram summarizing the variables and their associated probabilities. (c) What percent of voters are Independent but not swing voters? (d) What percent of voters are Independent or swing voters? (e) What percent of voters are neither Independent nor swing voters? (f) Is the event that someone is a swing voter independent of the event that someone is a political Independent?

Step by step solution

01

Determine If Events Are Mutually Exclusive

Two events are disjoint or mutually exclusive if they cannot occur at the same time. Here, being Independent and being a swing voter are not disjoint because 11% of respondents identified as both. This means some respondents are both politically Independent and swing voters.

02

Draw a Venn Diagram

Draw two overlapping circles, one representing Independents (35%) and the other representing swing voters (23%). In the overlapping region, which represents respondents who are both Independent and swing voters, place 11%. The part of the Independent circle not overlapping is 35% - 11% = 24%, and for swing voters 23% - 11% = 12%.

03

Calculate Percent Independent but Not Swing Voter

The part of the circle for Independents that does not overlap with swing voters is 35% (Independents) - 11% (both Independent and swing voters) = 24%. Thus, 24% of voters are Independent but not swing voters.

04

Calculate Percent Independent or Swing Voter

Use the formula for the union of two sets: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \). Here, \( P(A) = 0.35 \), \( P(B) = 0.23 \), and \( P(A \cap B) = 0.11 \). Therefore, \( P(A \cup B) = 0.35 + 0.23 - 0.11 = 0.47 \). Thus, 47% of voters are either Independent or swing voters.

05

Calculate Percent Neither Independent nor Swing Voter

Since 47% are either Independent or swing voters, 100% - 47% = 53% of voters are neither Independent nor swing voters.

06

Check for Independence of Events

Two events, \( A \) and \( B \), are independent if \( P(A \cap B) = P(A) \times P(B) \). Here, \( P(\text{Independent}) = 0.35 \), \( P(\text{Swing}) = 0.23 \), \( P(\text{Independent} \cap \text{Swing}) = 0.11 \). The product \( P(A) \times P(B) = 0.35 \times 0.23 = 0.0805 \), which is not equal to 0.11. Therefore, these events are not independent.

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Key Concepts

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

Mutually Exclusive Events

In probability theory, mutually exclusive events are those that cannot happen at the same time. Imagine having two groups that share no overlap. For this particular exercise, we need to determine if being an Independent voter and being a swing voter are mutually exclusive.
Since 11% of the respondents identified as both Independent voters and swing voters, this overlap indicates that these events are not mutually exclusive. If they were mutually exclusive, it would be impossible for someone to be in both groups simultaneously.
This example shows how to recognize non-mutually exclusive events in probability calculations by observing intersections in sets.

Venn Diagram

A Venn diagram is a helpful tool for visualizing the relationship between different groups. In probability, it allows us to show overlaps and calculate probabilities accurately. In this problem, we use two circles to represent the groups: one for Independents and the other for swing voters.
The circle for Independents represents 35%, and the swing voter circle represents 23%. The intersection of these circles, which illustrates individuals who are both Independent and swing voters, holds 11%.
This visualization helps us easily see that there are respondents who fit both categories and enables simple calculations for finding members exclusive to one group.

Independence of Events

Events are considered independent if the occurrence of one event does not affect the probability of the other. Mathematically, two events A and B are independent if the probability of both events occurring is equal to the product of their individual probabilities: \( P(A \cap B) = P(A) \times P(B) \).
In this exercise, Independence probability is 0.35, swing voter probability is 0.23, and the probability that someone is both is 0.11. Calculating the product gives 0.0805, not equal to 0.11.
Thus, identifying as an Independent voter does impact the likelihood of being a swing voter, showing these events aren't independent.

Probability Calculation

Understanding probability calculations is crucial for answering questions like those in this exercise. Let's discuss how to find probabilities for these groups based on given data.
First, to calculate the percentage of voters who are Independent but not swing voters, subtract the overlap (11%) from Independents (35%), giving us 24%.
Next, to find voters who fall into either category, use the union formula: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \). Substituting the values, we get 0.35 + 0.23 - 0.11 = 0.47 or 47%.
Finally, the complement of this gives us the probability of voters who fall into neither category: 100% - 47% = 53%. This exercise enhances our skills in practical probability calculations.