Step 1: Define the Random Variable

Let $X$ be the random variable representing the number of boys in a family with three children. The possible values of $X$ are 0, 1, 2, and 3.

Step 2: List Possible Outcomes

The sample space for the genders of three children is: {BBB, BBG, BGB, GBB, GGB, GBG, BGG, GGG}. Since the probability of having a boy or a girl is equal (0.5), each outcome in the sample space is equally likely.

Step 3: Calculate Probabilities

We need to calculate the probability of each possible value of $X$:

Step 4: Probability of 0 Boys (GGG)

There is only one outcome with 0 boys: GGG. $$P(X=0) = \frac{1}{8}$$

Step 5: Probability of 1 Boy (GGB, GBG, BGG)

There are three outcomes with 1 boy: GGB, GBG, BGG. $$P(X=1) = \frac{3}{8}$$

Step 6: Probability of 2 Boys (BBG, BGB, GBB)

There are three outcomes with 2 boys: BBG, BGB, GBB. $$P(X=2) = \frac{3}{8}$$

Step 7: Probability of 3 Boys (BBB)

There is only one outcome with 3 boys: BBB. $$P(X=3) = \frac{1}{8}$$

Step 8: Write the Probability Distribution

The probability distribution of $X$ is:

$X$ | 0 | 1 | 2 | 3 ------- | -------- | -------- | -------- | -------- $P(X)$ | 1/8 | 3/8 | 3/8 | 1/8