Expected number of pairs of consecutive sixes in after $n$ throws of a dice

Question

Suppose we throw a six-faced dice $n$ times.

Let's define the random variable X as the number of times two consecutive sixes appear, so $$ X = | \{i \in \{1, \ldots, n-1\} : r_i = r_{i+1} = 6\} | $$

where $r_i$ is the $i$th result in our sequence of throws.

How do I calculate $E[X]$?

Answer 1

Hint: Let $I_k$ be the indicator variable that both $k$ and $k+1$ are 6's.

Clearly $E[I_k] = \frac{1}{36}$.
Show that $ X = \sum_{k=1}^{n-1} I_k$.
Hence, by the linearity of expectation, $ E[X] = E [ \sum_{k=1}^{n-1} I_k ] = \sum_{k=1}^{n-1} E[I_k] = \frac{n-1}{36}$.