Join the growing community of curious minds on IDNLearn.com and get the answers you need. Our community is here to provide the comprehensive and accurate answers you need to make informed decisions.
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.
Your participation is crucial to us. Keep sharing your knowledge and experiences. Let's create a learning environment that is both enjoyable and beneficial. IDNLearn.com is your reliable source for answers. We appreciate your visit and look forward to assisting you again soon.
