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

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.

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

Accepted Answer

Welcome back, everyone. A couple plans to have 5 children. Assuming each child is equally likely to be a boy or a girl. What is the probability that at least 1 child is a girl? Let's define a random variable X which represents the number of girls. We want to identify the probability that X is greater than or equal to 1. And now using the addition rule we can list all of the possible outcomes. This will be the probability that X is exactly one. Plus, because we can have 4, the probability that access 2 plus the probability that access 3. Plus the probability that X is 4. And finally, because we have a total of 5 children, we're going to add the probability that all children are girls, the probability that X is 5. So, now, what we can see is that we have to identify 5 probabilities and this can be a lengthy process. So, instead, what we can do is identify the probability that X is greater than or equal to 1 using the method of complements. It is equal to 1 minus the probability of an opposite event, meaning that X is 0 and 0 only, right? Because this is the only possible outcome for an opposite event. So now what we're going to do is identify the probability that there would be 0 girls among 5 children, and this means that all 5 are boys. Now, the probability lowercase b of a boy or a girl is 1/2 because these probabilities are equally likely, so we divide 1 by 2. And what we have to do is simply use the multiplication rule for independent events. So we take one half and because we want to have 5 boys, we basically raise it to the power of 5. This gives us 1 divided by 32. This is the probability that all 5 children are boys. And now what we're going to do is substitute this result into the previous formula so that the probability of X being greater than or equal to 1 is 1 minus 1 divided by 32, which is 31 divided by 32. And that will be our final answer to this problem. Thank you for watching.

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

Key Concepts

Probability

Complement Rule

Binomial Distribution

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