Conditional probability formulas

Question

For a compound event $E_1E_2$,
Pr($E_1E_2$) = Pr($E_1$)Pr($E_2$|$E_1$)
such that if $E_1$ and $E_2$ are independent events, we can say that:
Pr($E_1E_2$) = Pr($E_1$)Pr($E_2$)

Is there an analogous formula that can be derived for the conditional probability: Pr($E_1E_2$|$E_3$)?

I would like to be able to reduce Pr($E_1E_2$|$E_3$) to terms that do not involve all three of the events $E_1$, $E_2$, $E_3$, given the assumption that only $E_1$ and $E_3$ are independent.

If this cannot be done, then what assumptions would be necessary in order to be able to do it?

Answer 1

Here is one way to write it given the assumption: $$ P(E_1\cap E_2\mid E_3)=P(E_2\mid E_1\cap E_3)P(E_1\mid E_3)=P(E_2\mid E_1\cap E_3)P(E_1). $$

Answer 2

Hint: If you write out $$Pr(E2|E1)=\frac{Pr(E_2\cap E_1)}{Pr(E_1)}$$ then you can investigate more complicated formulas involving more events. Don't remember a formula but understand them.