Conditional probability Old Kiwi - Rhea

Definition:

A conditional probability is the probability of one proposition given that another proposition holds. For the probability of proposition A given proposition B, we write P(A|B).

Formula

$ P(A|B) = \frac{P(A \cap B)}{P(B)} $

The probability of A given B is equal to the probability of A and B divided by the probability of B. Look at the example below; because we know B is true, we are only comparing the probabilities in the first row, $ P(A \cap B) $ and $ P(\lnot A \cap B) $. Therefore, we must divide the probability we are looking for, $ P(A \cap B) $, by the sum of all probabilities in the first row, P(B).

Example:

Assume the following probability distribution for propositions A and B (not A and not B are written as !A and !B respectively):

	A	!A
B	0.08	0.23
!B	0.12	0.57

In this example, the probability of A given B, P(A|B), is ~0.258 (0.08 / 0.31). Without knowing B, we would only be able to state the probability of A, P(A), as 0.2 (0.08 + 0.12).