How can I calculate the conditional probability of several events?

Question

Could you inform me please, how can I calculate conditioned probability of several events?

for example:

P (A | B, C, D) - ?

I know, that:

P (A | B) = P (A intersection B) / P (B)

But, unfortunately, I can't find any formula if an event A depends on several variables. Thanks in advance.

Answer 1

Take the intersection of B,C and D call it U. Then perform P(A|U).

Answer 2

Another approach would be:

P(A| B, C, D) = P(A, B, C, D)/P(B, C, D)
              = P(B| A, C, D).P(A, C, D)/P(B, C, D)
              = P(B| A, C, D).P(C| A, D).P(A, D)/{P(C| B, D).P(B, D)}
              = P(B| A, C, D).P(C| A, D).P(D| A).P(A)/{P(C| B, D).P(D| B).P(B)}

Note the similarity to:

      P(A| B) = P(A, B)/P(B)
              = P(B| A).P(A)/P(B)

And there are many equivalent forms.

Taking U = (B, C, D) gives: P(A| B, C, D) = P(A, U)/P(U)

P(A| B, C, D) = P(A, U)/P(U)
              = P(U| A).P(A)/P(U)
              = P(B, C, D| A).P(A)/P(B, C, D)

I'm sure they're equivalent, but do you want the joint probability of B, C & D given A?

Answer 3

check this wikipedia page under the sub-section named extensions, they do show how to derive conditional probability involving more than 2 events.