Expected value using indicator variable

Question

Suppose that 8ドル$ white balls and 2ドル$ black balls will be randomly ordered, from left to right (with all permutations of the 10ドル$ balls equally likely), what is the expected value of the number of balls that will be between the two black balls ?

I gave it a shot and this is where I am stuck.

If $E$ denotes the event that at least one one white ball in between the two black balls, then

$P(E) = 1- P(E^c)$

$P(E) = 1- \frac{(2) (9!)}{10!}$

which gives $P(E)$ to be equal to 0ドル.8$

I am not sure how to define the Indicator function with this information, that would lead me to answer the question.

Answer 1

Asociate an indicator variable ,$X_i$, to each white ball such that they take value 1ドル$ if it lies between the two black balls. Then with this formulation, the number of balls between the two black balls will be equal to the sum of $X_i$.

Hence, to solve the problem, we just have to consider the probability that a white ball is between the two black balls. For each white ball, it has 3ドル$ equally likely option. It can be on the left of both black balls, on the right of both black balls, or between the two black balls.

Hence, $$E\left[\sum_{i=1}^8X_i\right]=\sum_{i=1}^8P(X_i=1)=\frac83$$