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Sagot :
To find the expected value of the points Harlene gets from tossing two number cubes, we need to consider all possible outcomes and their respective probabilities.
### Step-by-Step Solution:
1. Calculate the Total Number of Outcomes:
When tossing two six-sided number cubes, each cube has 6 faces. Therefore, the total number of possible outcomes is:
[tex]\[ 6 \times 6 = 36 \][/tex]
2. Identify the Winning Outcomes:
Harlene wins 9 points if the sum of the two cubes is either 8 or 12. We need to find how many pairs of numbers (outcomes) result in these sums:
- Sum of 8:
- (2, 6)
- (3, 5)
- (4, 4)
- (5, 3)
- (6, 2)
There are 5 outcomes that result in a sum of 8.
- Sum of 12:
- (6, 6)
There is 1 outcome that results in a sum of 12.
Thus, the total number of winning outcomes is:
[tex]\[ 5 + 1 = 6 \][/tex]
3. Identify the Losing Outcomes:
Since the total number of outcomes is 36 and there are 6 winning outcomes, the number of losing outcomes is:
[tex]\[ 36 - 6 = 30 \][/tex]
4. Calculate the Probabilities:
- The probability of winning (getting a sum of 8 or 12) is:
[tex]\[ \frac{6}{36} = \frac{1}{6} \approx 0.1667 \][/tex]
- The probability of losing (not getting a sum of 8 or 12) is:
[tex]\[ \frac{30}{36} = \frac{5}{6} \approx 0.8333 \][/tex]
5. Calculate the Expected Value (EV):
The expected value (EV) is calculated by multiplying the points by their respective probabilities and then adding the results:
[tex]\[ \text{EV} = (9 \text{ points} \times \text{Probability of winning}) + (-2 \text{ points} \times \text{Probability of losing}) \][/tex]
[tex]\[ \text{EV} = (9 \times \frac{1}{6}) + (-2 \times \frac{5}{6}) \][/tex]
[tex]\[ \text{EV} = 1.5 + (-1.\overline{6}) = -0.1667 \][/tex]
6. Determine the Correct Answer:
Given the computed expected value of approximately -0.1667, the closest answer in the provided options is:
[tex]\[ -\frac{1}{6} \][/tex]
Thus, the expected value of the number of points for one roll is:
[tex]\[ -\frac{1}{6} \][/tex]
Consequently, the correct answer is [tex]\( -\frac{1}{6} \)[/tex].
### Step-by-Step Solution:
1. Calculate the Total Number of Outcomes:
When tossing two six-sided number cubes, each cube has 6 faces. Therefore, the total number of possible outcomes is:
[tex]\[ 6 \times 6 = 36 \][/tex]
2. Identify the Winning Outcomes:
Harlene wins 9 points if the sum of the two cubes is either 8 or 12. We need to find how many pairs of numbers (outcomes) result in these sums:
- Sum of 8:
- (2, 6)
- (3, 5)
- (4, 4)
- (5, 3)
- (6, 2)
There are 5 outcomes that result in a sum of 8.
- Sum of 12:
- (6, 6)
There is 1 outcome that results in a sum of 12.
Thus, the total number of winning outcomes is:
[tex]\[ 5 + 1 = 6 \][/tex]
3. Identify the Losing Outcomes:
Since the total number of outcomes is 36 and there are 6 winning outcomes, the number of losing outcomes is:
[tex]\[ 36 - 6 = 30 \][/tex]
4. Calculate the Probabilities:
- The probability of winning (getting a sum of 8 or 12) is:
[tex]\[ \frac{6}{36} = \frac{1}{6} \approx 0.1667 \][/tex]
- The probability of losing (not getting a sum of 8 or 12) is:
[tex]\[ \frac{30}{36} = \frac{5}{6} \approx 0.8333 \][/tex]
5. Calculate the Expected Value (EV):
The expected value (EV) is calculated by multiplying the points by their respective probabilities and then adding the results:
[tex]\[ \text{EV} = (9 \text{ points} \times \text{Probability of winning}) + (-2 \text{ points} \times \text{Probability of losing}) \][/tex]
[tex]\[ \text{EV} = (9 \times \frac{1}{6}) + (-2 \times \frac{5}{6}) \][/tex]
[tex]\[ \text{EV} = 1.5 + (-1.\overline{6}) = -0.1667 \][/tex]
6. Determine the Correct Answer:
Given the computed expected value of approximately -0.1667, the closest answer in the provided options is:
[tex]\[ -\frac{1}{6} \][/tex]
Thus, the expected value of the number of points for one roll is:
[tex]\[ -\frac{1}{6} \][/tex]
Consequently, the correct answer is [tex]\( -\frac{1}{6} \)[/tex].
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