To determine the expected value of the number of points Harlene will receive for one roll, we'll break down the question into several steps:
1. Identify all possible outcomes and desired outcomes:
- Each number cube has 6 faces, resulting in 36 (6 * 6) possible outcomes when two dice are rolled.
2. Calculate the number of ways to get sums of 8 and 12:
- Sum of 8: The possible combinations to get this sum are: (2,6), (3,5), (4,4), (5,3), and (6,2). This gives us 5 combinations.
- Sum of 12: The only way to get this sum is (6,6), which accounts for 1 combination.
- Therefore, there are a total of 5 (sum of 8) + 1 (sum of 12) = 6 ways to get either a sum of 8 or 12.
3. Determine the probabilities:
- Probability of winning (sum of 8 or 12):
[tex]\[
\frac{6}{36} = \frac{1}{6} \approx 0.16666666666666666
\][/tex]
- Probability of losing (any other sum):
Since there are 36 possible outcomes and only 6 of those are wins, the rest are losses:
[tex]\[
1 - \frac{1}{6} = \frac{5}{6} \approx 0.8333333333333334
\][/tex]
4. Calculate the points:
- Points for winning (getting a sum of 8 or 12): +9 points
- Points for losing (any other outcome): -2 points
5. Compute the expected value:
To calculate the expected value (E), we use the formula:
[tex]\[
E = (\text{Points for Win} \times \text{Probability of Win}) + (\text{Points for Loss} \times \text{Probability of Loss})
\][/tex]
Plugging in the numbers from above:
[tex]\[
E = (9 \times \frac{1}{6}) + (-2 \times \frac{5}{6})
\][/tex]
Simplify each term:
[tex]\[
E = 1.5 - 1.6666666666666667 \approx -0.16666666666666674
\][/tex]
The final expected value of the points for one roll is approximately [tex]\(-0.16666666666666674\)[/tex], which corresponds to the decimal form of [tex]\(-\frac{1}{6}\)[/tex].
Hence, the correct choice is:
[tex]\[
-\frac{1}{6}
\][/tex]
