Boy or Girl? In Exercises 71-74, a couple plans to have three chi...

4. Probability

Basic Concepts of Probability

Statistics

4. Probability

Basic Concepts of Probability

2:07 minutes

Problem 3.1.71

Textbook Question

Boy or Girl? In Exercises 71-74, a couple plans to have three children. Each child is equally likely to be a boy or a girl.
71. What is the probability that all three children are girls?

Verified step by step guidance

Step 1: Understand the problem. The couple plans to have three children, and each child is equally likely to be a boy or a girl. This means the probability of having a girl for each child is 0.5 (or 50%). We are tasked with finding the probability that all three children are girls.

Step 2: Recall the multiplication rule for independent events. Since the gender of each child is independent of the others, the probability of all three children being girls is the product of the probabilities of each child being a girl.

Step 3: Write the formula for the probability of all three children being girls. This can be expressed as: \( P(\text{all girls}) = P(\text{girl}) \times P(\text{girl}) \times P(\text{girl}) \).

Step 4: Substitute the probability of having a girl (0.5) into the formula. This gives: \( P(\text{all girls}) = 0.5 \times 0.5 \times 0.5 \).

Step 5: Simplify the expression to find the probability. Multiply the probabilities together: \( P(\text{all girls}) = 0.5^3 \). This represents the probability that all three children are girls.

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Key Concepts

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

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, the probability of each child being a girl is 0.5, since there are two equally likely outcomes: boy or girl. Understanding how to calculate probabilities is essential for solving questions related to random events.

Independent Events

Independent events are those whose outcomes do not affect each other. In this scenario, the gender of each child is independent of the others, meaning the outcome of one child's gender does not influence the others. This concept is crucial for calculating the overall probability of multiple events occurring together.

Multiplication Rule of Probability

The multiplication rule of probability states that the probability of multiple independent events occurring together is the product of their individual probabilities. For this question, to find the probability that all three children are girls, you would multiply the probability of each child being a girl (0.5) three times, resulting in (0.5) × (0.5) × (0.5).

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Boy or Girl? In Exercises 71-74, a couple plans to have three children. Each child is equally likely to be a boy or a girl.71. What is the probability that all three children are girls?