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On February 28, 2013, a Field Poll Survey reported that 61% of California registered voters approved of allowing two people of the same gender to marry and have regular marriage laws apply to them. Among 18 to 39 year olds (California registered voters), the approval rating was 78%. Six in ten California registered voters said that the upcoming Supreme Court’s ruling about the constitutionality of California’s Proposition 8 was either very or somewhat important to them. Out of those CA registered voters who support same-sex marriage, 75% say the ruling is important to them. In this problem, let: • C = California registered voters who support same-sex marriage. • B = California registered voters who say the Supreme Court’s ruling about the constitutionality of California’s Proposition 8 is very or somewhat important to them • A = California registered voters who are 18 to 39 years old. a. Find P(C). b. Find P(B). c. Find P(C|A). d. Find P(B|C). e. In words, what is C|A? f. In words, what is B|C? g. Find P(C AND B). h. In words, what is C AND B? i. Find P(C OR B). j. Are C and B mutually exclusive events? Show why or why not.

Step by step solution

01

Finding P(C)

The problem states that 61% of California registered voters support same-sex marriage. Therefore, \( P(C) = 0.61 \).

02

Finding P(B)

The problem indicates that six in ten California registered voters consider the Supreme Court's ruling on Proposition 8 to be important. Thus, \( P(B) = 0.60 \).

03

Finding P(C|A)

Among voters aged 18 to 39, 78% approve of same-sex marriage. Hence, \( P(C|A) = 0.78 \).

04

Finding P(B|C)

We are told that 75% of those who support same-sex marriage consider the ruling important. Therefore, \( P(B|C) = 0.75 \).

05

Understanding C|A

\( C|A \) describes the probability of voters supporting same-sex marriage given they are in the 18 to 39 age range.

06

Understanding B|C

\( B|C \) outlines the probability that voters consider the Supreme Court ruling important, given they support same-sex marriage.

07

Finding P(C AND B)

Using the formula \( P(C \cap B) = P(B|C) \times P(C) \), we get: \[ P(C \cap B) = 0.75 \times 0.61 = 0.4575 \].

08

Understanding C AND B

\( C \cap B \) is the event where a voter both supports same-sex marriage and finds the Supreme Court ruling important.

09

Finding P(C OR B)

Utilizing the formula \( P(C \cup B) = P(C) + P(B) - P(C \cap B) \), calculate: \[ P(C \cup B) = 0.61 + 0.60 - 0.4575 = 0.7525 \].

10

Checking for Mutual Exclusivity

Events C and B are not mutually exclusive since \( P(C \cap B) eq 0 \) (0.4575). This indicates some voters participate in both events.

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Key Concepts

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

Conditional Probability

Conditional probability is a fascinating concept in probability theory that helps us understand how likely an event is to occur, given that another event has already taken place. It's a valuable tool in statistical analysis to predict outcomes and make informed decisions.

For instance, let's consider the probability of a registered California voter approving same-sex marriage, given that they belong to the 18-39 age bracket. This is represented by the notation \( P(C|A) \). According to the problem statement, this probability is 0.78. In simpler terms, there is a 78% chance that a Californian voter approves of same-sex marriage if they fall within this younger demographic.

This kind of probability is crucial because it factors in existing conditions or events, especially when dealing with intricate data sets in surveys or studies. Understanding conditional probability helps in targeting specific audience groups based on their characteristics or tendencies, thereby enhancing decision-making processes and predicting trends.

Statistical Analysis

Statistical analysis is the process of gathering and analyzing data to uncover patterns and trends. It's a backbone of research and decision-making in many fields, including sociology, economics, and politics. By using statistics, we can make more informed predictions and decisions.

In the context of the given exercise, we apply statistical measures to interpret the opinions of California's registered voters on same-sex marriage and the importance of the Supreme Court's ruling on Prop 8. For example, calculating simple probabilities, such as \( P(C) = 0.61 \), reveals that 61% of voters support same-sex marriage. Similarly, \( P(B) = 0.60 \) highlights that 60% consider the ruling important. These numbers depict the support and concerns within the population being studied.

Statistical analysis also involves combining probabilities, as in the case of finding \( P(C \cap B) \) or \( P(C \cup B) \), to provide insights into how these events relate and overlap. This analysis helps in understanding public sentiment and can be used by policymakers to make data-driven decisions.

Mutual Exclusivity

When talking about mutual exclusivity in probability, it refers to two events that cannot happen at the same time. If events A and B are mutually exclusive, the occurrence of one event means the other cannot occur. This concept is fundamental in understanding relationships between different possible outcomes.

In our exercise, we are examining whether the events "Supporting same-sex marriage (C)" and "Considering the Supreme Court ruling important (B)" are mutually exclusive. To determine this, we look at the probability of both events happening, denoted as \( P(C \cap B) \). If \( P(C \cap B) \) equals zero, it implies mutual exclusivity, meaning no voter supports both events simultaneously.

However, as calculated, \( P(C \cap B) = 0.4575 \), which indicates that there are indeed voters who support same-sex marriage and also consider the ruling important. Thus, these events are not mutually exclusive. Understanding such relationships helps clarify how different segments of the population might overlap in their opinions or behaviors.