Q:

joint probability multiple events from single conditional events

I have five independent events $E_1,E_2,E_3,E_4, E_5$, with their conditional probability values for a given outcome.
Eg - $P(Y|E_1), P(Y|E_2), P(Y|E_3), P(Y|E_4), P(Y|E_5)$
Given the above can I find $P(Y|E_1, E_2), P(Y|E_4, E_3, E_5)$? which is basically any combination of the above events given as the condition. It does not have to be exact even an approximation is sufficient.

A:

No.   You need much more information about the series of $\{E_k\}$ events and their relation to $Y$.
For instant, since we have the following, we can see that we cannot express $\mathsf P(Y\mid E_1, E_2)$ in terms of $\mathsf P(Y\mid E_1),\mathsf P(Y\mid E_2)$ with just the given details.
$$\mathsf P(Y\mid E_1, E_2) =\frac{\mathsf P(Y, E_1\mid E_2)}{\mathsf P(E_1\mid E_2)} = \frac{\mathsf P(Y, E_1, E_2)}{\mathsf P(E_1, E_2)}$$

