Harlene tosses two number cubes. If a sum of 8 or 12 comes up, she gets 9 points. If not, she loses 2 points. What is the expected value of the number of points for one roll?

A. [tex]\(-\frac{2}{3}\)[/tex]
B. [tex]\(-\frac{1}{6}\)[/tex]
C. [tex]\(\frac{1}{6}\)[/tex]
D. [tex]\(\frac{2}{3}\)[/tex]



Answer :

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].