Answer:
The probability distribution is shown below.
Step-by-step explanation:
The urn consists of 8 white (<em>W</em>), 4 black (<em>B</em>) and 2 orange (<em>O</em>) balls.
The winning and losing criteria are:
- Win $2 for each black ball selected.
- Lose $1 for each white ball selected.
There are 8 + 4 + 2 = 14 balls in the urn.
The number of ways to select two balls is, 
ways.
The distribution of amount won or lost is as follows:
Outcomes: WW WO WB BB BO OO
X: -2 -1 1 4 2 0
Compute the probability of selecting 2 white balls as follows:
The number of ways to select 2 white balls is, 
ways.
The probability of WW is,
Compute the probability of selecting 1 white ball and 1 orange ball as follows:
The number of ways to select 1 white ball and 1 orange ball is, 
ways.
The probability of WO is,
Compute the probability of selecting 1 white ball and 1 black ball as follows:
The number of ways to select 1 white ball and 1 black ball is, 
ways.
The probability of WB is,
Compute the probability of selecting 2 black balls as follows:
The number of ways to select 2 black balls is, 
ways.
The probability of BB is,
Compute the probability of selecting 1 black ball and 1 orange ball as follows:
The number of ways to select 1 black ball and 1 orange ball is, 
ways.
The probability of BO is,
Compute the probability of selecting 2 orange balls as follows:
The number of ways to select 2 orange balls is, 
ways.
The probability of OO is,
The probability distribution of <em>X</em> is:
Outcomes: WW WO WB BB BO OO
X: -2 -1 1 4 2 0
P (X): 0.3077 0.1758 0.3516 0.0659 0.0879 0.0110
