Births in the United States In the United States, the true probab...

4. Probability

Multiplication Rule: Independent Events

Statistics

4. Probability

Multiplication Rule: Independent Events

2:03 minutes

Problem 4.3.7b

Textbook Question

Births in the United States In the United States, the true probability of a baby being a boy is 0.512 (based on the data available at this writing). For a family having three children, find the following.

b. The probability that all three children are boys.

Verified step by step guidance

Step 1: Understand the problem. We are tasked with finding the probability that all three children in a family are boys, given that the probability of a single child being a boy is 0.512.

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 boys is the product of the probabilities of each child being a boy.

Step 3: Write the formula for the probability of all three children being boys: \( P(\text{all boys}) = P(\text{boy}) \times P(\text{boy}) \times P(\text{boy}) \).

Step 4: Substitute the given probability of a single child being a boy (0.512) into the formula: \( P(\text{all boys}) = 0.512 \times 0.512 \times 0.512 \).

Step 5: Multiply the probabilities together to find the final result. This will give you the probability that all three children are boys.

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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 a baby being a boy is given as 0.512, indicating that there is a 51.2% chance of a boy being born. Understanding probability is essential for calculating the chances of multiple independent events occurring together.

Independent Events

Independent events are those whose outcomes do not affect each other. In the case of having three children, the gender of each child is independent of the others. This means that the probability of each child being a boy remains constant at 0.512, regardless of the genders of the other children.

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 boys, you would multiply the probability of having a boy (0.512) by itself three times, reflecting the independence of each child's gender.

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Births in the United States In the United States, the true probability of a baby being a boy is 0.512 (based on the data available at this writing). For a family having three children, find the following.b. The probability that all three children are boys.