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Product rule of probability

Synonyms: Multiplication rule of probability

Product rule of probability applies to the coincidence of two events (A and B)(A \land B).

In set theory symbols this is equivalent to the event C the intersection of events A and B:
C = A \cap B

The following special cases can be distinguished.

Mutually independent events

Two sets of events are mutually independent if we have
P(B) = P(B \mid A)).

That is the probability of event B if A occurred is identical to the probability B without the occurrence of A.

For two mutually independent events the probabilities can be multiplied.

P(A \cap B) = P(A) \cdot P(B)

Identical events

Two sets of events are identical if the intersection and union of these tow sets are identical.

A \cap B = A \cup B = A = B

For two identical events the probability is equal to each of the events.

P(A \cap B) = P(A) = P(B)

Not mutually independent events

This is the general case from which all the special cases above can be deduces.

P(A) \ne P(A \mid B)

For not mutually independent events we can write:

P(A \cap B) = P(A) \cdot P(B \mid A))

Tags: Statistics

Categories: Mathematics


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