Birdlover1234

The law of total probability is a theorem that states, in its discrete case, if $\left\{{B_{n}:n=1,2,3,\ldots }\right\}$ is a finite or countably infinite partition of a sample space (in other words, a set of pairwise disjoint events whose union is the entire sample space) and each event $B_{n}$ is measurable, then for any event $A$ of the same sample space:

$P(A)=\sum _{n}P(A\cap B_{n})$

or, alternatively,

$P(A)=\sum _{n}P(A\mid B_{n})P(B_{n}),$

where for any $n$ for which $P(B_{n})=0$ these terms are simply omitted from the summation since $P(A\mid B_{n})$ is finite.