Proof:Law of total probability

If , , , partition the sample space, then for any ,

Proof: Law of total probability

The proof is pretty clear from the following picture:

Since the [math]A_i [/math] are disjoint, we have that the sets are disjoint; and since every element of [math]S [/math] is in one of the [math]A_i [/math], we have that every element of [math]B [/math] is in one of the .

Therefore, we can apply the third Kolmogorov axiom to conclude

as required.