Using the sample in a family of four children, construct a probability distribution for the variable Z representing the number of boys. Draw the histogram of the probability distribution.

There are $2^4=16$ possible outcomes.

$P(Z=0)=P(GGGG)=(\dfrac{1}{2})^4=\dfrac{1}{16}$

$P(Z=1)=P(GGGB)+P(GGBG)$

$+P(GBGG)+P(BGGG)=4(\dfrac{1}{2})^4=\dfrac{1}{4}$

$P(Z=2)=P(GGBB)+P(GBBG)$

$+P(GBGB)+P(BGGB)+P(BGBG)$

$+P(BBGG)=6(\dfrac{1}{2})^4=\dfrac{3}{8}$

$P(Z=3)=P(GBBB)+P(BGBB)$

$+P(BBGB)+P(BBBG)=4(\dfrac{1}{2})^4=\dfrac{1}{4}$

$P(Z=4)=P(BBBB)=(\dfrac{1}{2})^4=\dfrac{1}{16}$

$\def\arraystretch{1.5} \begin{array}{c:c} z & 0 & 1 & 2 & 3 & 4 \\ \hline p(z) & 1/16 & 1/4 & 3/8 & 1/4 & 1/16 \\ \end{array}$