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Chapter 3: Problem 8

A couple has 2 children. What is the probability that both are girls if the older of the two is a girl?

Short Answer

Expert verified
The probability that both children are girls given that the older child is a girl is \( \frac{1}{2} \), or 50%.

Step by step solution

01

Identify the events

Let's denote the following events: A - the event that the older child is a girl B - the event that both children are girls We want to find the probability P(B|A), which is the probability that both children are girls given that the older child is a girl.
02

List the possible outcomes

There are 4 possible outcomes when a couple has two children, consisting of the gender of each child (girl or boy) as shown below: 1. Older child: Girl, Younger child: Girl (GG) 2. Older child: Girl, Younger child: Boy (GB) 3. Older child: Boy, Younger child: Girl (BG) 4. Older child: Boy, Younger child: Boy (BB) In our problem, we know that the older child is a girl, so we can eliminate outcomes 3 and 4. This leaves us with only 2 possible outcomes: 1. Older child: Girl, Younger child: Girl (GG) 2. Older child: Girl, Younger child: Boy (GB)
03

Calculate the conditional probability

Now that we have listed the possible outcomes, we can calculate the conditional probability P(B|A) using the following formula: \[ P(B|A) = \frac{P(A \cap B)}{P(A)} \] Now, we need to determine P(A) and P(A 鈭� B) to calculate the probability we are looking for. P(A) is the probability that the older child is a girl. Since there are 2 possibilities, GG and GB, each with an equal chance, P(A) = 1/2. P(A 鈭� B) is the probability that both events A and B occur together. In other words, it is the probability that the older child is a girl and both children are girls. There is only 1 outcome that satisfies this requirement, which is GG. So, P(A 鈭� B) = 1/4.
04

Calculate the final probability

Now that we have determined P(A) and P(A 鈭� B), we can find P(B|A) by substituting the values into the formula: \[ P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{1/4}{1/2} = \frac{1}{2} \] So, the probability that both children are girls given that the older child is a girl is 1/2, or 50%.

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