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Ch. 8 - Sequences, Induction, and Probability
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 8 - Sequences, Induction, and ProbabilityProblem 97
Chapter 9, Problem 97

What is the probability of a family having five boys born in a row? (Assume the probability of a male birth is 1/2.)

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Step 1: Understand the problem. The probability of a male birth is given as 1/2, and we are tasked with finding the probability of having five boys born consecutively. This involves multiplying probabilities for independent events.
Step 2: Recognize that each birth is an independent event. The probability of one boy being born does not affect the probability of the next boy being born.
Step 3: Write the probability for one boy being born as \( P(B) = \frac{1}{2} \). Since the events are independent, the probability of five boys being born consecutively is the product of the probabilities for each birth.
Step 4: Multiply the probabilities for five independent events: \( P(5 \text{ boys}) = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \).
Step 5: Simplify the expression. The result will be \( \left( \frac{1}{2} \right)^5 \), which represents the probability of having five boys born in a row.

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Key Concepts

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

Probability Basics

Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. In this context, the probability of a single event, such as the birth of a boy, is given as 1/2. Understanding basic probability principles is essential for calculating the likelihood of multiple independent events occurring in sequence.
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Independent Events

Independent events are those whose outcomes do not affect each other. In this scenario, the birth of each child is independent of the others, meaning the outcome of one birth does not influence the next. This concept is crucial for calculating the overall probability of multiple births resulting in boys, as it allows us to multiply the probabilities of each individual event.
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Multiplication Rule of Probability

The multiplication rule states that the probability of multiple independent events occurring together is the product of their individual probabilities. For this question, to find the probability of having five boys in a row, we multiply the probability of having a boy (1/2) by itself five times, resulting in (1/2)^5. This rule is fundamental for solving problems involving sequences of independent events.
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