Conditional Probability
A conditional probability is the likelihood of an event happening given that another event happens.
Description
This is the probability of an event given that some event(s) has (have) already occurred. This is generally notated as P(A|B) or P(A;B), where B has already occurred.
It is generically decomposed as P(A|B) = P(A∩B) / P(B). Importantly though, Bayes theorem provides the following decomposition based on joint probabilities:
Independence
If two events are independent (notated as A⫫B), then probabilities of one do not change from being conditioned on the other.
Put simply, if the conditioning probability is not 0, then:
P(A|B) = P(A)
P(B|A) = P(B)
A conditioning probability of 0 will cause the conditional probability to be undefined.
Conditional Independence
If events A and B are conditionally independent, then:
P(A|B,C) = P(A|C)
P(A,B|C) = P(A|C) P(B|C)
This interrelation is sometimes notated as (A⫫B)|C.