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As a project for math class, two students devised a game in which 3 black marbles and 2 red marbles are put into a bag. First, the players must decide who is playing black marbles and who is playing red marbles. Then each player takes a turn at 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 all aspects 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) will be twice that of [tex]$E$[/tex] (black), so Seth should choose to play red marbles.

C. Since [tex]$E($[/tex] red [tex]$)=E($[/tex] black $), 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 not play.



Answer :

GinnyAnswer
To determine which statement would be the best advice for Seth, let’s examine the expected values (E) for each color choice in the context of the game rules.

1. Points Scheme Recap:
- Both black marbles: +2 points
- Both red marbles: +4 points
- Different colors: -1 point
- No points for the opposite color pairs (no points for black marbles when both are red, and no points for red marbles when both are black).

2. Probabilities:
- Probability of drawing a black marble: [tex]\( \frac{3}{5} \)[/tex]
- Probability of drawing a red marble: [tex]\( \frac{2}{5} \)[/tex]

3. Expected Value Calculation:

For Black Marbles:
- Probability of both marbles being black: [tex]\( \left(\frac{3}{5}\right) \times \left(\frac{3}{5}\right) = \frac{9}{25} \)[/tex]
- Probability of drawing different colors: [tex]\( 2 \times \left(\frac{3}{5}\right) \times \left(\frac{2}{5}\right) = 2 \times \frac{6}{25} = \frac{12}{25} \)[/tex]
- Probability of both marbles being red (no points for black): [tex]\( \left(\frac{2}{5}\right) \times \left(\frac{2}{5}\right) = \frac{4}{25} \)[/tex]

Expected value for black marbles [tex]\( E_{\text{black}} \)[/tex] is calculated as:
[tex]\[ E_{\text{black}} = \left(\frac{9}{25} \times 2\right) + \left(\frac{12}{25} \times (-1)\right) = \left(\frac{18}{25}\right) - \left(\frac{12}{25}\right) = \frac{6}{25} = 0.24 \][/tex]

For Red Marbles:
- Probability of both marbles being red: [tex]\( \left(\frac{2}{5}\right) \times \left(\frac{2}{5}\right) = \frac{4}{25} \)[/tex]
- Probability of drawing different colors: [tex]\( 2 \times \left(\frac{2}{5}\right) \times \left(\frac{3}{5}\right) = 2 \times \frac{6}{25} = \frac{12}{25} \)[/tex]
- Probability of both marbles being black (no points for red): [tex]\( \left(\frac{3}{5}\right) \times \left(\frac{3}{5}\right) = \frac{9}{25} \)[/tex]

Expected value for red marbles [tex]\( E_{\text{red}} \)[/tex] is calculated as:
[tex]\[ E_{\text{red}} = \left(\frac{4}{25} \times 4\right) + \left(\frac{12}{25} \times (-1)\right) = \left(\frac{16}{25}\right) - \left(\frac{12}{25}\right) = \frac{4}{25} = 0.16 \][/tex]

4. Decision Making:
- It can be clearly seen that [tex]\( E_{\text{black}} = 0.24 \)[/tex] and [tex]\( E_{\text{red}} = 0.16 \)[/tex].
- Since the expected value for playing with black marbles (0.24) is higher than that for playing with red marbles (0.16), Seth should choose to play black marbles for the highest expected gain.

Conclusion:
The correct statement in all aspects guiding Seth to the optimal choice would be:
"Since [tex]\( E_{\text{black}} = 0.24 \)[/tex] and [tex]\( E_{\text{red}} = 0.16 \)[/tex], Seth should choose to play black marbles."

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