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Conditional Probability

In many situations, once more information becomes available; we are able to revise our estimates for the probability of further outcomes or events happening. For example, suppose you go out for lunch at the same place and time every Friday and you are served lunch within 15 minutes with probability 0.9. However, given that you notice that the restaurant is exceptionally busy, the probability of being served lunch within 15 minutes may reduce to 0.7. This is the conditional probability of being served lunch within 15 minutes given that the restaurant is exceptionally busy.

Let A and B be the two events associated with a random experiment. Then, the probability of occurrence of A under the condition that B has already occurred and P (B) ≠ 0 is called conditional probability and it is denoted by P (A|B).

Thus, P (A|B) = probability of occurrence of A given that B has already happened.

Similarly, P (B|A) = probability of occurrence of B given that A has already happened.

Sometimes, P (A|B) is also used to denote the probability occurrence of A when B occurs.

Similarly, P (B|A) is used to denote the probability occurrence of B when A occurs.

The usual notation for event A occurs given that event B has occurred is A|B (A given B). The symbol / is a vertical line and does not imply division. P (A|B) denotes the probability that event A will occur given that event B has occurred already.

A rule that can be used to determine a conditional probability from unconditional probabilities is:

Where:

P ( A ∩ B ) = the (unconditional) probability that event A and event B both occur

P(B) = the (unconditional) probability that event B occurs Baye's theorem shows the relation between one conditional probability and its inverse; for example, the probability of a hypothesis given observed evidence and the probability of that evidence given the hypothesis.

Baye's gave a special case involving continuous prior and posterior probability distributions and discrete probability distribution of data, but in its simplest setting involving only discrete distributions, Bayes theorem relates the conditional and marginal probabilities of events A and B, where B has a non-vanishing probability:

P(A|B)=(P(B|A) P(A)) / P(B)

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