Let $\{A\_n\}\_{n=1}^\infty$ be a sequence of events. 

If we let $I\_n$ be the indicator r.v. such that $I\_n=1$ when $A\_n$ occurs and $I\_n=0$ if $A\_n$ does not occur. 

I know that $P(A\_n) = E(I\_n)$

But why is it that $P(A\_j \cap A\_k) = E(I\_j I\_k)$? 

Further: If we define $\eta\_n = \sum\_{k=1}^n I\_k$, then 

$\sum\_{j=1}^n\sum\_{k=1}^n P(A\_j \cap A\_k) = E(\eta\_n^n)$

I don't understand this statement either, can someone please help