4. Probability
Multiplication Rule: Independent Events
1:46 minutesProblem 3.2.24a
Textbook Question
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)
a. Find the probability that all four have lost a friend or relative to murder.
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with finding the probability that all four randomly selected children have lost a friend or relative to murder. This involves using the Multiplication Rule for independent events, as the probability of one child being affected does not influence the others.
Step 2: Identify the given information. From the problem, we know that 8% of children (or 0.08 as a decimal) have lost a friend or relative to murder. This is the probability of a single child being affected, denoted as P(A) = 0.08.
Step 3: Recognize that the events are independent. Since the selection of one child does not affect the selection of another, the probability of all four children being affected can be calculated by multiplying the probabilities of each child being affected.
Step 4: Apply the Multiplication Rule. The probability of all four children being affected is given by the formula: P(All 4) = P(A) × P(A) × P(A) × P(A), which can also be written as P(All 4) = P(A)^4.
Step 5: Substitute the given probability into the formula. Replace P(A) with 0.08 in the formula P(All 4) = P(A)^4 to calculate the probability. This will give you the final result.
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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 occurring together is the product of their individual probabilities. In this context, if the events are the selection of children who have lost a friend or relative to murder, the rule allows us to calculate the probability of all selected children experiencing this event by multiplying the individual probabilities for each child.
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Independent Events
Independent events are those whose outcomes do not affect each other. In this scenario, the selection of one child does not influence the selection of another, assuming the sample size is large enough. This independence is crucial for applying the Multiplication Rule correctly, as it ensures that the probabilities can be multiplied without adjustment.
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Probability Calculation
Probability is a measure of the likelihood of an event occurring, expressed as a number between 0 and 1. To find the probability that all four selected children have lost a friend or relative to murder, we first determine the probability for one child (8% or 0.08) and then raise this probability to the power of four, reflecting the four independent selections.
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