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Sagot :
To determine the expected value of the game, follow these steps:
1. List the possible outcomes:
When rolling two six-sided dice, there are [tex]\(6 \times 6 = 36\)[/tex] possible outcomes. These outcomes range from sums of [tex]\(2\)[/tex] to [tex]\(12\)[/tex].
2. Identify win and lose sums:
- You win [tex]\(\$5\)[/tex] if the sum is [tex]\(2, 3, 4, 10, 11,\)[/tex] or [tex]\(12\)[/tex].
- You lose [tex]\(\$5\)[/tex] if the sum is [tex]\(5, 6, 7, 8,\)[/tex] or [tex]\(9\)[/tex].
3. Count the number of winning outcomes:
- Sum of 2: (1, 1) → 1 outcome
- Sum of 3: (1, 2) and (2, 1) → 2 outcomes
- Sum of 4: (1, 3), (2, 2), and (3, 1) → 3 outcomes
- Sum of 10: (4, 6), (5, 5), and (6, 4) → 3 outcomes
- Sum of 11: (5, 6) and (6, 5) → 2 outcomes
- Sum of 12: (6, 6) → 1 outcome
So, the number of winning outcomes is [tex]\(1 + 2 + 3 + 3 + 2 + 1 = 12\)[/tex].
4. Count the number of losing outcomes:
- Sum of 5: (1, 4), (2, 3), (3, 2), and (4, 1) → 4 outcomes
- Sum of 6: (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1) → 5 outcomes
- Sum of 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1) → 6 outcomes
- Sum of 8: (2, 6), (3, 5), (4, 4), (5, 3), and (6, 2) → 5 outcomes
- Sum of 9: (3, 6), (4, 5), (5, 4), and (6, 3) → 4 outcomes
So, the number of losing outcomes is [tex]\(4 + 5 + 6 + 5 + 4 = 24\)[/tex].
5. Calculate probabilities:
- Probability of winning (P(win)) = Number of winning outcomes / Total outcomes = [tex]\(12 / 36 = 1/3 \approx 0.333\)[/tex]
- Probability of losing (P(lose)) = Number of losing outcomes / Total outcomes = [tex]\(24 / 36 = 2/3 \approx 0.667\)[/tex]
6. Calculate the expected value:
The expected value (EV) is calculated using the formula:
[tex]\[ \text{EV} = (\text{P(win)} \times \text{win amount}) + (\text{P(lose)} \times \text{lose amount}) \][/tex]
Here, the win amount is [tex]\(\$5\)[/tex] and the lose amount is [tex]\(-\$5\)[/tex]. Substituting the probabilities we get:
[tex]\[ \text{EV} = (0.333 \times 5) + (0.667 \times -5) = 1.665 - 3.335 = -1.67 \][/tex]
Therefore, the expected value of the game is [tex]\(-\$1.67\)[/tex]. This means, on average, you would expect to lose [tex]\(\$1.67\)[/tex] per game in the long run.
1. List the possible outcomes:
When rolling two six-sided dice, there are [tex]\(6 \times 6 = 36\)[/tex] possible outcomes. These outcomes range from sums of [tex]\(2\)[/tex] to [tex]\(12\)[/tex].
2. Identify win and lose sums:
- You win [tex]\(\$5\)[/tex] if the sum is [tex]\(2, 3, 4, 10, 11,\)[/tex] or [tex]\(12\)[/tex].
- You lose [tex]\(\$5\)[/tex] if the sum is [tex]\(5, 6, 7, 8,\)[/tex] or [tex]\(9\)[/tex].
3. Count the number of winning outcomes:
- Sum of 2: (1, 1) → 1 outcome
- Sum of 3: (1, 2) and (2, 1) → 2 outcomes
- Sum of 4: (1, 3), (2, 2), and (3, 1) → 3 outcomes
- Sum of 10: (4, 6), (5, 5), and (6, 4) → 3 outcomes
- Sum of 11: (5, 6) and (6, 5) → 2 outcomes
- Sum of 12: (6, 6) → 1 outcome
So, the number of winning outcomes is [tex]\(1 + 2 + 3 + 3 + 2 + 1 = 12\)[/tex].
4. Count the number of losing outcomes:
- Sum of 5: (1, 4), (2, 3), (3, 2), and (4, 1) → 4 outcomes
- Sum of 6: (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1) → 5 outcomes
- Sum of 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1) → 6 outcomes
- Sum of 8: (2, 6), (3, 5), (4, 4), (5, 3), and (6, 2) → 5 outcomes
- Sum of 9: (3, 6), (4, 5), (5, 4), and (6, 3) → 4 outcomes
So, the number of losing outcomes is [tex]\(4 + 5 + 6 + 5 + 4 = 24\)[/tex].
5. Calculate probabilities:
- Probability of winning (P(win)) = Number of winning outcomes / Total outcomes = [tex]\(12 / 36 = 1/3 \approx 0.333\)[/tex]
- Probability of losing (P(lose)) = Number of losing outcomes / Total outcomes = [tex]\(24 / 36 = 2/3 \approx 0.667\)[/tex]
6. Calculate the expected value:
The expected value (EV) is calculated using the formula:
[tex]\[ \text{EV} = (\text{P(win)} \times \text{win amount}) + (\text{P(lose)} \times \text{lose amount}) \][/tex]
Here, the win amount is [tex]\(\$5\)[/tex] and the lose amount is [tex]\(-\$5\)[/tex]. Substituting the probabilities we get:
[tex]\[ \text{EV} = (0.333 \times 5) + (0.667 \times -5) = 1.665 - 3.335 = -1.67 \][/tex]
Therefore, the expected value of the game is [tex]\(-\$1.67\)[/tex]. This means, on average, you would expect to lose [tex]\(\$1.67\)[/tex] per game in the long run.
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