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.
74. What is the probability that at least one child is a boy?

Verified step by step guidance

1. 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. We are tasked with finding the probability that at least one child is a boy.
2. Step 2: Use the complement rule. Instead of directly calculating the probability of 'at least one boy,' calculate the probability of the complement event, which is 'no boys' (all children are girls), and subtract it from 1. This is because P(at least one boy) = 1 - P(no boys).
3. Step 3: Calculate the probability of the complement event (all children are girls). Since each child is equally likely to be a boy or a girl, the probability of a girl for one child is 0.5. For three children, the probability of all being girls is the product of their individual probabilities: P(all girls) = 0.5 × 0.5 × 0.5.
4. Step 4: Subtract the complement probability from 1 to find the desired probability. Use the formula P(at least one boy) = 1 - P(all girls).
5. Step 5: Simplify the expression to get the final probability. Ensure that the result is expressed as a fraction, decimal, or percentage, depending on the context of the problem.

Key Concepts

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, it helps us quantify the chances of having at least one boy among three children, where each child has an equal chance of being a boy or a girl.

Complement Rule

The Complement Rule states that the probability of an event occurring is equal to 1 minus the probability of it not occurring. For this question, instead of directly calculating the probability of having at least one boy, we can find the probability of having no boys (all girls) and subtract it from 1.

Binomial Distribution

The Binomial Distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this scenario, the number of boys in three children can be modeled using this distribution, where 'success' is defined as having a boy.