Question

ASSESSMENT: Construct the probability distribution of the situation below:

Two balls are drawn in succession without replacement from an urn 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 B.

Answer 1

The possible values that $B$ can take are $0,1,$ and $2.$

Each of these numbers corresponds to an event in the sample space $S=\{ww, wb, bw, bb\}$ of equally likely outcomes for this experiment:

$B=0$ to $\{ww\},$ $B=1$ to $\{wb, bw\},$ $B=2$ to $\{bb\}.$

The probability of each of these events, hence of the corresponding value of $B,$ can be found simply by counting, to give

$B=0:$

$$P(B)=\dfrac{5}{5+6}\cdot\dfrac{5-1}{5+6-1}=\dfrac{2}{11}$$

$B=1:$

$$P(B)=\dfrac{5}{5+6}\cdot\dfrac{6}{5+6-1}+\dfrac{6}{5+6}\cdot\dfrac{5}{5+6-1}=\dfrac{6}{11}$$

$B=2:$

$$P(B)=\dfrac{6}{5+6}\cdot\dfrac{6-1}{5+6-1}=\dfrac{3}{11}$$

$$\def\arraystretch{1.5} \begin{array}{c:c} B & 0 & 1 & 2 \\ \hline \\ P(B) &\dfrac{2}{11} & \dfrac{6}{11} & \dfrac{3}{11} \\ \end{array}$$

This table is the probability distribution of $B.$