# Conditional probability given multiple independent events

I am interested in finding the conditional probability $P(A|E_1,E_2,...,E_n)$ where the $E_i$ are mutually independent events. I know only $P(A)$ and $P(A|E_i)$.

Is this possible? If so, how? If not, what information is missing?

## 1 Answer

\begin{align} \mathsf P(A\mid \bigcap_i E_i) & = \frac{\mathsf P(\bigcap_i E_i\mid A)\cdot\mathsf P(A)}{P(\bigcap_i E_i)} \\[1ex] & = \frac{\mathsf P(\bigcap_i E_i\mid A)\cdot \mathsf P(A)}{\prod_i \mathsf P(E_i)} \end{align}

To proceed further you require a guarantee of conditional independence.   That is: $\mathsf P(\bigcap_i E_i\mid A) = \prod_i\mathsf P(E_i\mid A)$

However, mutually independent events $\{E_1, E_2, \ldots E_n\}$ are not necessarily mutually, conditionally independent given event $A$.

• Conditional independence took me awhile to grok. An example helps: let P(G|I,D) be the probability that a student gets a grade of G = {A,B,C,D,F} on an exam conditioned on their intellegence I and the exam difficulty D. Clearly I and D are independent until you have the evidence of their grade G; Once you know the grade, I and D are no longer indepedent (i.e., if the student got an F and they are really smart, then you know something about how difficult the exam was). Aug 28, 2018 at 21:47
• Seems to be contradicting this answer : math.stackexchange.com/questions/568029/… Oct 30, 2018 at 9:36
• @AkshayLAradhya Indeed, it is. Can you see why? Do you agree? Oct 30, 2018 at 12:09