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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.
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.
We need to calculate the probability of each possible value of $X$:
There is only one outcome with 0 boys: GGG. $$P(X=0) = \frac{1}{8}$$
There are three outcomes with 1 boy: GGB, GBG, BGG. $$P(X=1) = \frac{3}{8}$$
There are three outcomes with 2 boys: BBG, BGB, GBB. $$P(X=2) = \frac{3}{8}$$
There is only one outcome with 3 boys: BBB. $$P(X=3) = \frac{1}{8}$$
The probability distribution of $X$ is:
$X$ | 0 | 1 | 2 | 3 ------- | -------- | -------- | -------- | -------- $P(X)$ | 1/8 | 3/8 | 3/8 | 1/8
Final Answer: X | 0 | 1 | 2 | 3 ------- | -------- | -------- | -------- | -------- P(X) | 1/8 | 3/8 | 3/8 | 1/8