Final answer:
The variance of x, the number of rolls up to and including the 76th roll that comes up 1, is 9.
Step-by-step explanation:
The number of rolls up to and including the 76th roll that comes up 1 can be represented by a geometric distribution. In this case, since the die has 10 sides and only one of them shows a 1, the probability of rolling a 1 on any given roll is 1/10. The expected value of a geometric distribution is 1/p, where p is the probability of success. So, the expected value of x is 10.
To calculate the variance of x, we can use the formula for the variance of a geometric distribution: Var(x) = (1-p)/p^2. In this case, p = 1/10, so Var(x) = (1-(1/10))/(1/10)^2 = 9/1 = 9.