Question

Two balls are drawn in succession without replacement of an containing 5 white balls and 6 black balls Let B be the random variable representing the number of black balls. Construct the probability distribution of the random variable R

Answer 1

Let $X=$ the random variable representing the number of black balls in result.

There are $5+6=11$ balls. Then

$$P(W)=5/11,P(B)=6/11$$

$$S=\{WW, WB, BW, BB\}$$

The posiible value of $X: 0, 1, 2.$

$$P(X=0)=P(WW)=\dfrac{\dbinom{6}{0}\dbinom{5}{2-0}}{\dbinom{11}{2}}$$

$$=\dfrac{1(10)}{55}=\dfrac{2}{11}$$

$$P(X=1)=P(WB)+P(BW)=\dfrac{\dbinom{6}{1}\dbinom{5}{2-1}}{\dbinom{11}{2}}$$

$$=\dfrac{6(5)}{55}=\dfrac{6}{11}$$

$$P(X=2)=P(BB)=\dfrac{\dbinom{6}{2}\dbinom{5}{2-2}}{\dbinom{11}{2}}$$

$$=\dfrac{15(1)}{55}=\dfrac{3}{11}$$

Construct the probability distribution of the random variable $X$

$$\begin{matrix} x & 0 & 1 & 2 \\ p(x) & 2/11 & 6/11 & 3/11 \end{matrix}$$