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?
-\\(\frac{2}{3}
-\\(\frac{1}{8}
\\(\frac{1}{8}
\\(\frac{2}{3})

Explanation:

Step1: Calculate total outcomes

When two number - cubes are tossed, the total number of outcomes is \(6\times6 = 36\).

Step2: Find number of ways to get sum of 8 or 12

For the sum of two dice to be 8: \((2,6),(3,5),(4,4),(5,3),(6,2)\) - 5 ways. For the sum of two dice to be 12: \((6,6)\) - 1 way. So, the number of favorable outcomes (sum is 8 or 12) is \(5 + 1=6\).

Step3: Calculate probabilities

The probability of getting a sum of 8 or 12, \(P(8\text{ or }12)=\frac{6}{36}=\frac{1}{6}\). The probability of not getting a sum of 8 or 12, \(P(\text{not }8\text{ or }12)=1 - \frac{1}{6}=\frac{5}{6}\).

Step4: Calculate expected value

The expected - value formula is \(E(X)=\sum_{i}x_ip_i\). Here, \(x_1 = 9\) (points when sum is 8 or 12) with probability \(p_1=\frac{1}{6}\), and \(x_2=- 2\) (points when sum is not 8 or 12) with probability \(p_2=\frac{5}{6}\). So, \(E(X)=9\times\frac{1}{6}+(-2)\times\frac{5}{6}=\frac{9 - 10}{6}=-\frac{1}{6}\).

Answer:

\(-\frac{1}{6}\)