Conditional Probability
A conditional probability is the likelihood of an event happening given that another event happens. The math notation is P(A|B), as in the probability of A given B.
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.