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Sagot :
Let's analyze the given problem step-by-step in a structured manner by calculating the expected values (E) for choosing black marbles and red marbles.
### Defining Probabilities
First, we define the probabilities of drawing each color of marble:
- Probability of drawing a black marble ([tex]\( P(\text{black}) \)[/tex]) is [tex]\(\frac{3}{5}\)[/tex].
- Probability of drawing a red marble ([tex]\( P(\text{red}) \)[/tex]) is [tex]\(\frac{2}{5}\)[/tex].
### Calculating the Expected Value for Black Marble Player
The expected value (E) for the player choosing black marbles considers all possible outcomes:
1. Drawing two black marbles (Both black): [tex]\( P(\text{black}) \times P(\text{black}) \)[/tex]
2. Drawing one black and one red marble (Different colors): [tex]\( 2 \times P(\text{black}) \times P(\text{red}) \)[/tex] (two ways)
#### Points Assigned to Outcomes for Black Marbles
- If both drawn marbles are black (Black, Black): +2 points
- If one marble is black and the other is red (Black, Red or Red, Black): -1 point
- If both are red (Red, Red): 0 points
#### Calculation:
[tex]\[ E(\text{black}) = (P(\text{black} \cap \text{black}) \times 2) + (P(\text{black} \cap \text{red}) \times -1) + (P(\text{red} \cap \text{red}) \times 0) \][/tex]
[tex]\[ E(\text{black}) = \left(\frac{3}{5} \cdot \frac{3}{5} \times 2\right) + \left(2 \cdot \frac{3}{5} \cdot \frac{2}{5} \times -1\right) + \left(\frac{2}{5} \cdot \frac{2}{5} \times 0\right) \][/tex]
### Calculating the Expected Value for Red Marble Player
The expected value (E) for the player choosing red marbles also considers all possible outcomes:
1. Drawing two red marbles (Both red): [tex]\( P(\text{red}) \times P(\text{red}) \)[/tex]
2. Drawing one red and one black marble (Different colors): [tex]\( 2 \times P(\text{red}) \times P(\text{black}) \)[/tex] (two ways)
#### Points Assigned to Outcomes for Red Marbles
- If both drawn marbles are red (Red, Red): +4 points
- If one marble is red and the other is black (Red, Black or Black, Red): -1 point
- If both are black (Black, Black): 0 points
#### Calculation:
[tex]\[ E(\text{red}) = (P(\text{red} \cap \text{red}) \times 4) + (P(\text{red} \cap \text{black}) \times -1) + (P(\text{black} \cap \text{black}) \times 0) \][/tex]
[tex]\[ E(\text{red}) = \left(\frac{2}{5} \cdot \frac{2}{5} \times 4\right) + \left(2 \cdot \frac{2}{5} \cdot \frac{3}{5} \times -1\right) + \left(\frac{3}{5} \cdot \frac{3}{5} \times 0\right) \][/tex]
After computing these values, we find:
- [tex]\( E(\text{black}) = 0.24 \)[/tex]
- [tex]\( E(\text{red}) = 0.16 \)[/tex]
### Conclusion
Since [tex]\( E(\text{black}) = 0.24 \)[/tex] and [tex]\( E(\text{red}) = 0.16 \)[/tex]:
- Seth should choose to play black marbles because the expected value for the person choosing black marbles is higher. Therefore, the correct statement helping Seth make the best choice is:
"Since [tex]\(E(\text{black}) = 0.24\)[/tex] and [tex]\(E(\text{red}) = 0.16\)[/tex], Seth should choose to play black marbles."
### Defining Probabilities
First, we define the probabilities of drawing each color of marble:
- Probability of drawing a black marble ([tex]\( P(\text{black}) \)[/tex]) is [tex]\(\frac{3}{5}\)[/tex].
- Probability of drawing a red marble ([tex]\( P(\text{red}) \)[/tex]) is [tex]\(\frac{2}{5}\)[/tex].
### Calculating the Expected Value for Black Marble Player
The expected value (E) for the player choosing black marbles considers all possible outcomes:
1. Drawing two black marbles (Both black): [tex]\( P(\text{black}) \times P(\text{black}) \)[/tex]
2. Drawing one black and one red marble (Different colors): [tex]\( 2 \times P(\text{black}) \times P(\text{red}) \)[/tex] (two ways)
#### Points Assigned to Outcomes for Black Marbles
- If both drawn marbles are black (Black, Black): +2 points
- If one marble is black and the other is red (Black, Red or Red, Black): -1 point
- If both are red (Red, Red): 0 points
#### Calculation:
[tex]\[ E(\text{black}) = (P(\text{black} \cap \text{black}) \times 2) + (P(\text{black} \cap \text{red}) \times -1) + (P(\text{red} \cap \text{red}) \times 0) \][/tex]
[tex]\[ E(\text{black}) = \left(\frac{3}{5} \cdot \frac{3}{5} \times 2\right) + \left(2 \cdot \frac{3}{5} \cdot \frac{2}{5} \times -1\right) + \left(\frac{2}{5} \cdot \frac{2}{5} \times 0\right) \][/tex]
### Calculating the Expected Value for Red Marble Player
The expected value (E) for the player choosing red marbles also considers all possible outcomes:
1. Drawing two red marbles (Both red): [tex]\( P(\text{red}) \times P(\text{red}) \)[/tex]
2. Drawing one red and one black marble (Different colors): [tex]\( 2 \times P(\text{red}) \times P(\text{black}) \)[/tex] (two ways)
#### Points Assigned to Outcomes for Red Marbles
- If both drawn marbles are red (Red, Red): +4 points
- If one marble is red and the other is black (Red, Black or Black, Red): -1 point
- If both are black (Black, Black): 0 points
#### Calculation:
[tex]\[ E(\text{red}) = (P(\text{red} \cap \text{red}) \times 4) + (P(\text{red} \cap \text{black}) \times -1) + (P(\text{black} \cap \text{black}) \times 0) \][/tex]
[tex]\[ E(\text{red}) = \left(\frac{2}{5} \cdot \frac{2}{5} \times 4\right) + \left(2 \cdot \frac{2}{5} \cdot \frac{3}{5} \times -1\right) + \left(\frac{3}{5} \cdot \frac{3}{5} \times 0\right) \][/tex]
After computing these values, we find:
- [tex]\( E(\text{black}) = 0.24 \)[/tex]
- [tex]\( E(\text{red}) = 0.16 \)[/tex]
### Conclusion
Since [tex]\( E(\text{black}) = 0.24 \)[/tex] and [tex]\( E(\text{red}) = 0.16 \)[/tex]:
- Seth should choose to play black marbles because the expected value for the person choosing black marbles is higher. Therefore, the correct statement helping Seth make the best choice is:
"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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