Question

A meeting of consuls was attended by 4 Americans and 2 Germans. If three consuls were selected at random one after the other, illustrate the probability distribution of a random variable G (number of Germans) and draw the histogram.

Answer 1

Let's denote G - Germans, A - Americans.

Sample space S is all possible outcomes.

$$S = \{AAA, AAG, AGA, GAA, AGG, GAG, GGA\}$$

Let $g$ be the random variables representing the number of Germans that occur. The possible values of $g$ are $0,1,2.$

$$\def\arraystretch{1.5} \begin{array}{c:c} Sample\ point & g \\ \hline AAA & 0 \\ \hdashline AAG & 1 \\ \hdashline AGA & 1 \\ \hdashline GAA & 1 \\ \hdashline AGG & 2 \\ \hdashline GAG & 2 \\ \hdashline GGA & 2 \\ \hdashline \end{array}$$

$$P(G=0)=\dfrac{4}{6}(\dfrac{3}{5})(\dfrac{2}{4})=\dfrac{1}{5}$$

$$P(G=1)=\dfrac{4}{6}(\dfrac{3}{5})(\dfrac{2}{4})+\dfrac{4}{6}(\dfrac{2}{5})(\dfrac{3}{4})+\dfrac{2}{6}(\dfrac{4}{5})(\dfrac{3}{4})=\dfrac{3}{5}$$

$$P(G=2)=\dfrac{4}{6}(\dfrac{2}{5})(\dfrac{1}{4})+\dfrac{2}{6}(\dfrac{4}{5})(\dfrac{1}{4})+\dfrac{2}{6}(\dfrac{1}{5})(\dfrac{4}{4})=\dfrac{1}{5}$$

The probability distribution of a random variable $G$

$$\def\arraystretch{1.5} \begin{array}{c:c} g & 0 & 1 & 2 \\ \hline p(g) & 1/5 & 3/5 & 1/5 \end{array}$$