Probability question and die rolling

Question

A die is rolled infinitely many times. I want to find

(a) The expected number of consecutive rolls of (6,6) among the first 1000 rolls. And the expected number of consecutive rolls of (1,2) among the first 1000 rolls.

(b) What is larger and by how much: the expected number of rolls till the first occurrence of (6,6) or the expected number of rolls till the first occurrence of (1,2)?

(c) What is the probability that (6,6) occurs before (1,2)?

Answer 1

I am answering question(a).

Let$X_n$ be the value of nth roll.The number of (6,6)'s in the first 1000 rolls is given by $\displaystyle\sum_{n=1}^{999}\mathbb{I}(X_n=6 and X_{n+1}=6)$

By linearity of expectation

E$[\displaystyle\sum_{n=1}^{999}\mathbb{I}(X_n=6 and X_{n+1}=6)]$

=$\displaystyle\sum_{n=1}^{999}E[\mathbb{I}(X_n=6 and X_{n+1}=6)]$

=$\displaystyle\sum_{n=1}^{999}\mathbb{P}(X_n=6 and X_{n+1}=6)$

=$\frac{999}{36}$

Similarly, the number of(1,2)'s in the first 1000 rolls is $\frac{999}{36}$