harlene tosses two number cubes. if a sum of 8 or 12 comes up, she gets 9 points. if not, she loses 2…

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

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

Answer

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})