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Home Math Background Probability Conditional Probability | |||||
See also: Bayesian rule, independent events | |||||
Conditional ProbabilityA conditional probability is defined as the probability of an event, given that another event has occurred. This means that the probability for event A is effected by event B. Formally, a conditional probability is depicted as P(A | B) (read: the probability of the event A under the condition that event B occurred).
The calculation of the conditional probability P(A | B) involves two steps. First we have to realize that the fact that event B occurred reduces the sample space, to the sample points of B. This applies to the whole sample space which is now B and also the sample space of A. Only the sample points of event A that also belong to event B can occur thereafter; these are the sample points of the intersection of A and B. Since the probability is the ratio of the number of sample points of an event to the total number of sample points, the conditional probability is: P(A | B) = P(A B) / P(B) This is true under the condition that P(B) is not equal to zero. The
equation adjusts the probability of A
B from its original value in the whole sample space to the probability
in the reduced sample space B.
Intersection of eventsThe probability of an intersection of events is calculated by the multiplicative rule, which makes use of conditional probabilities. We simply re-arrange the equation for the conditional probabilityP(B| A) = P(A B) / P(A) and obtain:P(A B) = P(A)P(B|A)
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