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Each player takes a turn drawing a marble, noting the color, replacing the marble in the bag, and then drawing a second marble and noting the color before returning it to the bag. The point scheme for the game is detailed in the table below.

\begin{tabular}{|l|l|}
\hline
\multicolumn{2}{|c|}{Point Values for Marble Game} \\
\hline
\multicolumn{1}{|c|}{Black Marble Points} & \multicolumn{1}{c|}{Red Marble Points} \\
\hline
Both black: +2 points & Both red: +4 points \\
\hline
Different colors: -1 point & Different colors: -1 point \\
\hline
Both red: 0 points & Both black: 0 points \\
\hline
\end{tabular}

If Seth is challenged to a game by a classmate, which statement below is correct in helping him make the correct choice?

A. Since [tex]$E($[/tex] black [tex]$) = 0.24$[/tex] and [tex]$E($[/tex] red [tex]$) = 0.16$[/tex], Seth should choose to play black marbles.

B. [tex]$E($[/tex] red [tex]$)$[/tex] will be twice that of [tex]$E($[/tex] black [tex]$)$[/tex], so Seth should choose to play red marbles.

C. Since [tex]$E($[/tex] red [tex]$) = E($[/tex] black [tex]$)$[/tex], it is a fair game, so it doesn't matter which color Seth chooses.

D. Both options will lose points because there are two ways to lose points and only one way to gain points. He should choose neither color.



Answer :

GinnyAnswer
Let's break down the problem step by step to determine the expected values for both black and red marbles and see which option Seth should choose, based on the expected points.

First, let’s summarize the information provided:

1. Probabilities:
- Probability of drawing a black marble (P_black) = 0.24
- Probability of drawing a red marble (P_red) = 0.16

2. Points for outcomes:
- Both marbles black: +2 points
- Both marbles red: +4 points
- One marble black and one marble red (different colors): -1 point

Now, we'll calculate the expected value of the points Seth can earn for each marble he plays. The expected value (E) of points is a way of predicting the average outcome of a random event, in this instance, selecting marbles of a certain color.

### Expected Value for Black Marbles:

When Seth plays with black marbles, the scenarios to consider are:
- Drawing two black marbles
- Drawing one black and one red marble

#### Calculation:
- Probability of both marbles black: P_black P_black
- Probability of one black and one red marble: P_black P_red

Then we multiply these probabilities by their respective points and sum them up to get the expected value for black marbles.

Expected points for two black marbles:
[tex]\[ (P\_black \times P\_black) \times \text{points\_both\_black} \][/tex]
[tex]\[ (0.24 \times 0.24) \times 2 = 0.1152 \][/tex]

Expected points for one black and one red marble:
[tex]\[ (P\_black \times P\_red) \times \text{points\_different\_colors} \][/tex]
[tex]\[ (0.24 \times 0.16) \times -1 = -0.0384 \][/tex]

Adding these together gives the expected value for selecting black marbles:
[tex]\[ E\_black = 0.1152 + (-0.0384) = 0.0768 \][/tex]

### Expected Value for Red Marbles:

When Seth plays with red marbles, the scenarios to consider are:
- Drawing two red marbles
- Drawing one red and one black marble

#### Calculation:
- Probability of both marbles red: P_red P_red
- Probability of one red and one black marble: P_red P_black

Then we multiply these probabilities by their respective points and sum them up to get the expected value for red marbles.

Expected points for two red marbles:
[tex]\[ (P\_red \times P\_red) \times \text{points\_both\_red} \][/tex]
[tex]\[ (0.16 \times 0.16) \times 4 = 0.1024 \][/tex]

Expected points for one red and one black marble:
[tex]\[ (P\_red \times P\_black) \times \text{points\_different\_colors} \][/tex]
[tex]\[ (0.16 \times 0.24) \times -1 = -0.0384 \][/tex]

Adding these together gives the expected value for selecting red marbles:
[tex]\[ E\_red = 0.1024 + (-0.0384) = 0.064 \][/tex]

### Conclusion

The expected value for black marbles is [tex]\(0.0768\)[/tex] and for red marbles is [tex]\(0.064\)[/tex]. Therefore, based on the expected value, Seth should choose black marbles since the expected value of playing with black marbles (0.0768) is higher than that of red marbles (0.064).

Thus, the correct statement is:
Seth should choose to play black marbles.