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Sagot :
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."
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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