7. a 12 - sided solid has faces numbered 1 through 12.\na. find the probability of rolling a number greater…

Explanation:

Step1: Calculate favorable outcomes for part a

The numbers greater than 5 on a 1 - 12 numbered solid are 6, 7, 8, 9, 10, 11, 12. So there are 7 favorable outcomes. The total number of outcomes is 12.

Step2: Calculate probability for part a

The probability formula is $P(A)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. So $P(\text{number}> 5)=\frac{7}{12}$.

Step3: Calculate favorable outcomes for part b

The numbers 3, 9, 11 are the favorable outcomes. So there are 3 favorable outcomes out of 12 total outcomes.

Step4: Calculate expected value for part b

The expected - value formula is $E = n\times P(A)$, where $n$ is the number of trials and $P(A)$ is the probability of the event. The probability $P(\text{3, 9 or 11})=\frac{3}{12}=\frac{1}{4}$, and $n = 180$. Then $E=180\times\frac{1}{4}=45$.

Answer:

a. $\frac{7}{12}$ b. 45