Question

a meeting of consuls was attended by 4 Americans and 4 Germans. If three consuls were selected at random, construct the probability distribution of the random variable G representing the number of Germans.

Answer 1

Let $A$ represent the number of Americans, $G$ represent the number of Germans.

We can select three consuls from $4+4=8$ consuls in $\dbinom{8}{3}=56$ ways.

The possible values of random variable $G$ representing the number of Germans are $0,1,2,3.$

$$P(G=0)=\dfrac{\dbinom{4}{0}\dbinom{4}{3-0}}{\dbinom{8}{3}}=\dfrac{1(4)}{56}=\dfrac{1}{14}$$

$$P(G=1)=\dfrac{\dbinom{4}{1}\dbinom{4}{3-1}}{\dbinom{8}{3}}=\dfrac{4(6)}{56}=\dfrac{3}{7}$$

$$P(G=2)=\dfrac{\dbinom{4}{2}\dbinom{4}{3-2}}{\dbinom{8}{3}}=\dfrac{6(4)}{56}=\dfrac{3}{7}$$

$$P(G=3)=\dfrac{\dbinom{4}{3}\dbinom{4}{3-3}}{\dbinom{8}{3}}=\dfrac{4(1)}{56}=\dfrac{1}{14}$$

The probability distribution of the random variable G representing the number of Germans is

$$\def\arraystretch{1.5} \begin{array}{c:c} g & 0 & 1 & 2 & 3 \\ \hline p(g) & 1/14 & 3/7 & 3/7 & 1/14 \end{array}$$