Let x determine a random variable, and use your knowledge of probability to prepare a probability distribution. A family has two children and the number of girls is recorded. (Assume an equal chance of a boy or girl for each birth.) Complete the probability distribution. x \( \mathrm{P}(\mathrm{x}) \) (Type an integer or a simplified fraction.)

1. A family with two children has four equally likely outcomes for the gender composition: - Boy-Boy - Boy-Girl - Girl-Boy - Girl-Girl 2. Define the random variable \( x \) as the number of girls in the family. 3. For each outcome, count the number of girls: - Boy-Boy: \( x = 0 \) - Boy-Girl: \( x = 1 \) - Girl-Boy: \( x = 1 \) - Girl-Girl: \( x = 2 \) 4. Each outcome has a probability of \(\frac{1}{4}\). 5. Now, construct the probability distribution: - For \( x = 0 \): \( P(0) = \frac{1}{4} \) - For \( x = 1 \): There are 2 outcomes giving one girl, so \( P(1) = \frac{2}{4} = \frac{1}{2} \) - For \( x = 2 \): \( P(2) = \frac{1}{4} \) 6. The completed probability distribution is: | \( x \) | \( P(x) \) | |---------|--------------| | 0 | \( \frac{1}{4} \) | | 1 | \( \frac{1}{2} \) | | 2 | \( \frac{1}{4} \) |