Chapter 4, Problem 4.1P

Two balls are chosen randomly from an urn containing 8 white, 4 black, and 2 orange balls. Suppose that we win $2 for each black ball selected and we lose $1 for each white ball selected. Let
X denote our winnings. What are the possible values of
X, and what are the probabilities associated with each value?

To determine

To conclude: The possible values of X and the probabilities associated with each value.

Answer to Problem 4.1P

x	1	2	3	4
p(x)	2/7	3/13	1/6	1/11

Explanation of Solution

Given:

In an urn, there are 8 white, 4 black and 2 orange balls. If player will select a black ball then he will win $2 and if not then he will lose $1 for each white ball selected.

Calculation:

Consider that B and W are defined as black and white.

Further, consider that there are 4 black balls.

Hence, X’s possible values are 1,2,3, and 4.

The probability that select 1,2,3 and 4 black ball, respectively is:

  $\begin{array}{c}P\left(X=1\right)=\frac{n\left(b\right)\left(\text{Number of available black balls}\right)}{n}\\ =\frac{4}{14}\\ =\frac{2}{7}\end{array}$

  $\begin{array}{c}P\left(X=2\right)=\frac{n\left(b\right)\left(\text{Number of available black balls}\right)}{n}\\ =\frac{3}{13}\\ =\frac{3}{13}\end{array}$

  $\begin{array}{c}P\left(X=3\right)=\frac{n\left(b\right)\left(\text{Number of available black balls}\right)}{n}\\ =\frac{2}{12}\\ =\frac{1}{6}\end{array}$

  $\begin{array}{c}P\left(X=4\right)=\frac{n\left(b\right)\left(\text{Number of available black balls}\right)}{n}\\ =\frac{1}{11}\end{array}$