"Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.
23. Celebrities as Role Models In a sample of 1103 probable voters, three out of four say they would like entertainers to address social and political issues. Two probable voters are selected at random. (Source: The Hollywood Reporter)
c. Find the probability that at least one of the two probable voters would like entertainers to address social and political issues."

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the probability that at least one of the two probable voters selected at random would like entertainers to address social and political issues. This involves using the complement rule and the multiplication rule.

Step 2: Define the probabilities. From the problem, we know that three out of four voters (or 75%) would like entertainers to address social and political issues. Thus, the probability of a voter liking this is P(A) = 0.75, and the probability of a voter not liking this is P(A') = 1 - 0.75 = 0.25.

Step 3: Use the complement rule. The probability of 'at least one' liking the issue is the complement of the probability that 'none' of the two voters like the issue. Mathematically, P(at least one) = 1 - P(none).

Step 4: Calculate P(none). To find P(none), use the multiplication rule. Since the two voters are selected independently, the probability that both do not like the issue is P(A') * P(A'). Substituting the values, P(none) = 0.25 * 0.25.

Step 5: Substitute into the complement formula. Finally, substitute P(none) into the complement formula: P(at least one) = 1 - P(none). This gives the probability that at least one of the two voters would like entertainers to address social and political issues.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Multiplication Rule

The Multiplication Rule in probability states that the probability of two independent events both occurring is the product of their individual probabilities. This rule is essential for calculating the likelihood of multiple outcomes happening together, especially when events do not influence each other.

Complementary Events

Complementary events are pairs of outcomes in a probability scenario where one event occurs if and only if the other does not. In this context, finding the probability that at least one of the two voters supports entertainers addressing social issues can be simplified by calculating the probability that neither does and subtracting it from one.

Probability Calculation

Probability calculation involves determining the likelihood of an event occurring, expressed as a number between 0 and 1. In this problem, the probability of a voter wanting entertainers to address issues is 0.75, and the probability of not wanting this is 0.25, which are crucial for applying the Multiplication Rule and finding the desired probability.

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