"Using the Multiplication Rule In Exercises 19-32, use the Multiplication Rule.
24. Knowing a Person Who Was Murdered In a sample of 11,771 children ages 2 to 17, 8% have lost a friend or relative to murder. Four children are selected at random. (Adapted from University of New Hampshire)
b. Find the probability that none of the four has lost a friend or relative to murder."

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the probability that none of the four randomly selected children has lost a friend or relative to murder. This involves using the Multiplication Rule for independent events.

Step 2: Define the probability of the complementary event. The problem states that 8% of children have lost a friend or relative to murder. Therefore, the probability that a child has NOT lost a friend or relative to murder is 1 - 0.08 = 0.92.

Step 3: Recognize that the four children are selected independently. This means the outcome for one child does not affect the outcomes for the others. The Multiplication Rule for independent events states that the probability of all events occurring is the product of their individual probabilities.

Step 4: Apply the Multiplication Rule. To find the probability that none of the four children has lost a friend or relative to murder, multiply the probability of the complementary event (0.92) by itself four times: \( P(\text{none}) = 0.92 \times 0.92 \times 0.92 \times 0.92 \).

Step 5: Simplify the expression. This can also be written as \( P(\text{none}) = 0.92^4 \). Use this formula to calculate the final probability if needed.

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Key Concepts

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

Multiplication Rule of Probability

The Multiplication Rule states that the probability of two independent events both occurring is the product of their individual probabilities. In this context, it helps calculate the likelihood of multiple children not having lost a friend or relative to murder by multiplying the probabilities of each child independently not experiencing this loss.

Complementary Probability

Complementary probability refers to the probability of an event not occurring. In this scenario, if 8% of children have lost someone to murder, then 92% have not. This concept is crucial for determining the probability that none of the selected children have experienced this loss, as it allows us to use the probability of the complementary event.

Independent Events

Independent events are those whose outcomes do not affect each other. In this problem, the selection of each child is independent, meaning the probability of one child not having lost someone to murder does not change based on the selections of the others. This independence is essential for applying the Multiplication Rule correctly.

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