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# multiplication law for probabilities

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Law for probabilities stating that if A and B are independent events then

P(AB)=P(A)×P(B),

and, in the case of n independent events, A1, A2,…, An,

P(A1A2 ∩…∩ An)=P(A1)×P(A2)×…×P(An).

This is a special case of the more general law of compound probability, which holds for events that may not be independent. In the case of two events, A and B, this law states that

P(AB)=P(A)×P(B|A)=P(B)×P(A|B).

For three events, A, B, and C, this becomes

P(ABC)=P(A)×P(B|A)×P(C|AB).

There are six (=3!) alternative right-hand sides, for example P(C)×P(A|C)×P(B|CA). The generalization to more than three events can be inferred. For definitions of symbols, see conditional probability; intersection.

Subjects: Probability and Statistics.

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