Laws of total probability  Probability
Laws of total probability
Lessons
Notes:
Recall:
P(A and B)=P(A)$\cdot$P(BA) or equivalently, P(A and B)=P(B)$\cdot$P(AB)
The Law of Total Probability:
P(A)=P(A and B)+P(A and ~B)=P(B)P(AB)+P(~B)P(A~B)
Or in full generality, if all of $B_1, B_2,...B_n$ include the entire sample space S, and are all pairwise mutually exclusive then:
$P(A)=P(A$ and $B_1)+P(A$ and $B_2)+ \cdots +P(A$ and $B_n)$
$=P(B_1)P(AB_1)+P(B_2)P(AB_2)+ \cdots + P(B_n)P(AB_n)$

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