Laws of total probability - Probability

Laws of total probability


P(A and B)=P(A)\cdotP(B|A) or equivalently, P(A and B)=P(B)\cdotP(A|B)

The Law of Total Probability:
P(A)=P(A and B)+P(A and ~B)=P(B)P(A|B)+P(~B)P(A|~B)

Or in full generality, if all of B1,B2,...BnB_1, B_2,...B_n include the entire sample space S, and are all pairwise mutually exclusive then:

P(A)=P(AP(A)=P(A and B1)+P(AB_1)+P(A and B2)++P(AB_2)+ \cdots +P(A and Bn)B_n)
=P(B1)P(AB1)+P(B2)P(AB2)++P(Bn)P(ABn)=P(B_1)P(A|B_1)+P(B_2)P(A|B_2)+ \cdots + P(B_n)P(A|B_n)
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Laws of total probability

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