Final answer:
The variance of the number of times a 3 is rolled up to the 80th roll on a 14-sided die is 20/49.
Step-by-step explanation:
The random variable x represents the number of times a 3 is rolled up to and including the 80th roll. Since this is a 14-sided die, the probability of rolling a 3 on each roll is 1/14. In order to find Var(x), we need to first calculate the expected value of x using the formula E(x) = np, where n is the number of rolls and p is the probability of success. In this case, n = 80 and p = 1/14. Therefore, E(x) = (80)(1/14) = 40/7.
To find Var(x), we can use the formula Var(x) = E(x^2) - [E(x)]^2. The expected value of x squared can be calculated as E(x^2) = (n)(p^2) + np - [E(x)]^2. Plugging in the values, Var(x) = (80)(1/14)^2 + (80)(1/14) - (40/7)^2.
Simplifying the expression, we get Var(x) = 20/49. Therefore, the answer is b) 20/49.