Question

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

Answer 1

There are $2^4=16$ possible outcomes.

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

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

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

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

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

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

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

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

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

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