Question

wo balls are chosen randomly from an um containing 8 white, 4 black,and 2 orange balls. Suppose that we win $2 for each black ball selected and we lose $1for each white ball selected. LetXdenote our winnings. What are the possible valuesofX, and what are the probabilities associated with each value

Answer 1

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, ${14\choose 2}=91$ 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, ${8\choose 2}=28$ ways.

The probability of WW is,

$P(WW)=\frac{n(WW)}{N}=\frac{28}{91}=0.3077$

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, ${8\choose 1}\times {2\choose 1}=16$ ways.

The probability of WO is,

$P(WO)=\frac{n(WO)}{N}=\frac{16}{91}=0.1758$

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, ${8\choose 1}\times {4\choose 1}=32$ ways.

The probability of WB is,

$P(WB)=\frac{n(WB)}{N}=\frac{32}{91}=0.3516$

Compute the probability of selecting 2 black balls as follows:

The number of ways to select 2 black balls is, ${4\choose 2}=6$ ways.

The probability of BB is,

$P(BB)=\frac{n(BB)}{N}=\frac{6}{91}=0.0659$

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, ${4\choose 1}\times {2\choose 1}=8$ ways.

The probability of BO is,

$P(BO)=\frac{n(BO)}{N}=\frac{8}{91}=0.0879$

Compute the probability of selecting 2 orange balls as follows:

The number of ways to select 2 orange balls is, ${2\choose 2}=1$ ways.

The probability of OO is,

$P(OO)=\frac{n(OO)}{N}=\frac{1}{91}=0.0110$

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