# Proof:Law of total probability

($1)$2 | $3 ($4) | $5 ($6)
If , , , partition the sample space, then for any ,

Proof: Law of total probability
The proof is pretty clear from the following picture:

Since the are disjoint, we have that the sets are disjoint; and since every element of is in one of the , we have that every element of is in one of the .

Therefore, we can apply the third Kolmogorov axiom to conclude

as required.