"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)
c. Find the probability that at least one of the four has lost a friend or relative to murder."

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with finding the probability that at least one of the four randomly selected children has lost a friend or relative to murder. This is a complementary probability problem, where we first calculate the probability that none of the four children has lost a friend or relative to murder, and then subtract this value from 1.
2. Step 2: Define the probability of the complementary event. The probability that a single child has NOT lost a friend or relative to murder is 1 - 0.08 = 0.92 (since 8% of children have lost someone to murder).
3. Step 3: Use the Multiplication Rule to calculate the probability that none of the four children has lost a friend or relative to murder. Assuming the selections are independent, the probability that all four children have NOT lost someone is given by the product of their individual probabilities: \( P(\text{none}) = 0.92 \times 0.92 \times 0.92 \times 0.92 = 0.92^4 \).
4. Step 4: Calculate the probability of the event we are interested in (at least one child has lost a friend or relative to murder). This is the complement of the probability that none of the children has lost someone: \( P(\text{at least one}) = 1 - P(\text{none}) = 1 - 0.92^4 \).
5. Step 5: Conclude the solution. The final probability can be found by evaluating \( 1 - 0.92^4 \). This gives the probability that at least one of the four children has lost a friend or relative to murder.

Key Concepts

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 events happening together, especially when dealing with random selections, as in this question.

Complement Rule

The Complement Rule is a fundamental concept in probability that states the probability of an event occurring is equal to one minus the probability of it not occurring. In this context, to find the probability that at least one child has lost a friend or relative to murder, it is often easier to first calculate the probability that none have, and then subtract that from one.

Binomial Probability

Binomial Probability refers to the probability of obtaining a fixed number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this scenario, selecting four children can be modeled as a binomial experiment where 'success' is defined as a child having lost a friend or relative to murder, allowing for the application of binomial probability formulas.