Basketball player indicator random variable question

Question

A certain basketball player practices shooting free throws over and over again. The shots are independent, with probability $ p $ of success. In $ n $ shots, what is the expected number of streaks of $ 7 $ consecutive successful shots? (Note that, for example, $ 9 $ in a row counts as $ 3 $ streaks.)

I do not understand why we cannot use the binomial distribution and answer $ 7 p $ here for the expected shots and then divide by $ 3 $, so the answer is $ \frac { 7 p } 3 $. The actual answer is $ ( n - 6 ) p ^ 7 $.

Answer 1

For $k = 7$ to $n$,
let $X_k$ be an indicator r.v that $= 1$ if there is a streak including it and the $6$ preceding numbers and $0$ otherwise.

Then $P(X_k) = p^7$

The expectation of an indicator variable is just the probability of the event it points to, so

$\Bbb E[X_k] = p^7$,

and by linearity of expectation, which holds even if the r.v's are not independent,

$\Bbb E[X] = \Bbb E[X_7] + \Bbb E[X_8] + \Bbb E[X_9] + ...\Bbb E[X_n] = (n-6)p^7$