Conditional Probability

Notation P(A|B), the probability of event A given that B has occurred.

Draw a ven diagram - find what outcomes in B are also in A. So you reduce the probability of A to the probability of A ∩ B.

P(A∩B)/P(B), P(B) > 0

Terms

P(A)   prior prob of A
P(A|B) posterior prob of A

Multiplication rule

P(A∩B) = P(B)P(A|B)

Bays Theorem

Let P(B) > 0. Then,

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

Law of total probability

Given two events A and B from the same sample space,

B = (B ∩ A) ∪ (B ∩ Acomplement)

P(B)
= P(B ∩ A) + P(B∩Acomplement)
= P(B|A)P(A) + P(B|Acomplement)P(Acomplement)

This can be extended to any n sets.

A1, A2,...,An where A1 ∩ ... ∩ An = {} and sum of all A's is S.

Then

P(B) = sum(1, n) P(B|Ak)P(Ak)

Other stuff

P(A union B) = P(A) + P(B) - P(A intersect B)