Respuesta :
Answer:
The probability distribution is shown below.
Step-by-step explanation:
The urn consists of 8 white (W), 4 black (B) and 2 orange (O) 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, [tex]{14\choose 2}=91[/tex] 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, [tex]{8\choose 2}=28[/tex] ways.
The probability of WW is,
[tex]P(WW)=\frac{n(WW)}{N}=\frac{28}{91}=0.3077[/tex]
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, [tex]{8\choose 1}\times {2\choose 1}=16[/tex] ways.
The probability of WO is,
[tex]P(WO)=\frac{n(WO)}{N}=\frac{16}{91}=0.1758[/tex]
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, [tex]{8\choose 1}\times {4\choose 1}=32[/tex] ways.
The probability of WB is,
[tex]P(WB)=\frac{n(WB)}{N}=\frac{32}{91}=0.3516[/tex]
Compute the probability of selecting 2 black balls as follows:
The number of ways to select 2 black balls is, [tex]{4\choose 2}=6[/tex] ways.
The probability of BB is,
[tex]P(BB)=\frac{n(BB)}{N}=\frac{6}{91}=0.0659[/tex]
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, [tex]{4\choose 1}\times {2\choose 1}=8[/tex] ways.
The probability of BO is,
[tex]P(BO)=\frac{n(BO)}{N}=\frac{8}{91}=0.0879[/tex]
Compute the probability of selecting 2 orange balls as follows:
The number of ways to select 2 orange balls is, [tex]{2\choose 2}=1[/tex] ways.
The probability of OO is,
[tex]P(OO)=\frac{n(OO)}{N}=\frac{1}{91}=0.0110[/tex]
The probability distribution of X 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
