To find the expected value of the number of points Harlene gets for one roll of two number cubes, we need to follow a series of steps:
1. Determine the possible outcomes and their respective probabilities:
There are a total of [tex]\(36\)[/tex] possible outcomes when rolling two number cubes (each cube has [tex]\(6\)[/tex] faces, thus [tex]\(6 \times 6 = 36\)[/tex] outcomes).
Let's identify the probabilities of getting a sum of [tex]\(8\)[/tex] or [tex]\(12\)[/tex]:
- Sum of [tex]\(8\)[/tex]: There are [tex]\(5\)[/tex] combinations that produce this sum [tex]\((2, 6), (3, 5), (4, 4), (5, 3), (6, 2)\)[/tex].
- Sum of [tex]\(12\)[/tex]: There is [tex]\(1\)[/tex] combination that produces this sum [tex]\((6, 6)\)[/tex].
The total number of combinations that result in a sum of [tex]\(8\)[/tex] or [tex]\(12\)[/tex] is:
[tex]\(5 + 1 = 6\)[/tex].
Therefore, the probability of rolling a sum of [tex]\(8\)[/tex] or [tex]\(12\)[/tex] is:
[tex]\[
\frac{6}{36} = \frac{1}{6}
\][/tex]
The probability of not rolling a sum of [tex]\(8\)[/tex] or [tex]\(12\)[/tex] is:
[tex]\[
1 - \frac{1}{6} = \frac{5}{6}
\][/tex]
2. Determine the points associated with each outcome:
- If a sum of [tex]\(8\)[/tex] or [tex]\(12\)[/tex] comes up, Harlene gets [tex]\(9\)[/tex] points.
- If the sum is anything else, Harlene loses [tex]\(2\)[/tex] points.
3. Calculate the expected value:
The expected value [tex]\(E\)[/tex] is calculated by multiplying each outcome's points by its probability and then summing these products:
[tex]\[
E = \left(\frac{1}{6} \times 9\right) + \left(\frac{5}{6} \times (-2)\right)
\][/tex]
Breaking it down:
- The expected contribution from rolling a sum of [tex]\(8\)[/tex] or [tex]\(12\)[/tex] is:
[tex]\[
\frac{1}{6} \times 9 = 1.5
\][/tex]
- The expected contribution from not rolling a sum of [tex]\(8\)[/tex] or [tex]\(12\)[/tex] is:
[tex]\[
\frac{5}{6} \times (-2) = -1.6666666666666667
\][/tex]
Summing these contributions yields the overall expected value:
[tex]\[
E = 1.5 + (-1.6666666666666667) = -0.16666666666666674
\][/tex]
Thus, the expected value of the number of points Harlene gets in one roll is:
[tex]\[
-\frac{1}{6}
\][/tex]
So the correct answer is:
[tex]\[
\boxed{-\frac{1}{6}}
\][/tex]
