Conditional probabilities

In probability theory, conditional probability is a measure of the probability of an event given that (by assumption, presumption, assertion or evidence) another event has occurred. If the event of interest is A and the event B is known or assumed to have occurred, "the conditional probability of A given B", or "the probability of A under the condition B", is usually written as P(A|B), or sometimes P_B(A). For example, the probability that any given person has a cough on any given day may be only 5%. But if we know or assume that the person has a cold, then they are much more likely to be coughing. The conditional probability of coughing given that you have a cold might be a much higher 75%.

The concept of conditional probability is one of the most fundamental and one of the most important concepts in probability theory. But conditional probabilities can be quite slippery and require careful interpretation. For example, there need not be a causal or temporal relationship between A and B.

P(A|B) may or may not be equal to P(A) (the unconditional probability of A). If P(A|B) = P(A) (or its equivalent P(B|A) = P(B)), then events A and B are said to be independent: in such a case, having knowledge about either event does not change our knowledge about the other event. Also, in general, P(A|B) (the conditional probability of A given B) is not equal to P(B|A). For example, if you have cancer you might have a 90% chance of testing positive for cancer. In this case what is being measured is that the if event B "having cancer" has occurred, the probability A - test is positive given that B having cancer occurred is 90%, P(A|B) = 90%. Alternatively, you can test positive for cancer but you may have only a 10% chance of actually having cancer because cancer is very rare. In this case what is being measured is the probability of the event B - having cancer given that the event A - test is positive has occurred, P(B|A) = 10%. Falsely equating the two probabilities causes various errors of reasoning such as the base rate fallacy. Conditional probabilities can be correctly reversed using Bayes' theorem.

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