Question

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?$\boldsymbol{-\frac{2}{3}}$$\boldsymbol{-\frac{1}{6}}$$\boldsymbol{\frac{1}{6}}$$\boldsymbol{\frac{2}{3}}$

Explanation:

Step1: Find total possible outcomes

When rolling two number cubes, each has 6 faces, so total outcomes: $6 \times 6 = 36$

Step2: Count favorable outcomes (sum 8 or 12)

• Sum of 8: (2,6), (3,5), (4,4), (5,3), (6,2) → 5 outcomes
• Sum of 12: (6,6) → 1 outcome

Total favorable outcomes: $5 + 1 = 6$

Step3: Calculate probabilities

Probability of winning 9 points: $\frac{6}{36} = \frac{1}{6}$
Probability of losing 2 points: $1 - \frac{1}{6} = \frac{5}{6}$

Step4: Compute expected value

Expected value = (Points for win × Win probability) + (Points for loss × Loss probability)
$\text{Expected Value} = 9 \times \frac{1}{6} + (-2) \times \frac{5}{6}$
$= \frac{9}{6} - \frac{10}{6}$
$= \frac{9 - 10}{6} = -\frac{1}{6}$

Answer:

$-\frac{1}{6}$ (Option B)