Compute conditional probabilities of dependent events

Question

We are given an outcome A and a three different events $E_{1}$, $E_{2}$ and $E_{3}$, $E_{4}$. All the events are mutually dependent.

Now, we would like to compute:

$P(A|E_1, E_2, E_3, E_4)$

Is there a way to compute this using joint probabilities? How would you express it using conditional probabilities?

Answer 1

By definition of conditional probability: $\mathsf P(A\mid E_1,E_2,E_3,E_4)=\dfrac{\mathsf P(A,E_1,E_2,E_3,E_4)}{\mathsf P(E_1,E_2,E_3,E_4)}$

And of course, $\mathsf P(E_1,E_2,E_3,E_4)=\mathsf P(A,E_1,E_2,E_3,E_4)+\mathsf P(A^\complement,E_1,E_2,E_3,E_4)$

And so on.