Total Probability Theorem

Theorem

Let $\left({\Omega, \Sigma, \Pr}\right)$ be a probability space.

Let $\left\{{B_1, B_2, \ldots}\right\}$ be a partition of $\Omega$ such that $\forall i: \Pr \left({B_i}\right) > 0$.

Then:

$\forall A \in \Sigma: \Pr \left({A}\right) = \sum_i \Pr \left({A \mid B_i}\right) \Pr \left({B_i}\right)$

Proof

	\(\displaystyle \)	\(\displaystyle \Pr \left({A}\right)\)	\(=\)	\(\displaystyle \Pr \left({A \cap \left({\bigcup_i B_i}\right)}\right)\)	\(\displaystyle \)	from Intersection with Subset is Subset, as $\bigcup_i B_i = \Omega$ and $A \subseteq \Omega$
	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \Pr \left({\bigcup_i \left({A \cap B_i}\right)}\right)\)	\(\displaystyle \)	Intersection Distributes over Union
	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sum_i \Pr \left({A \cap B_i}\right)\)	\(\displaystyle \)	as all the $A \cap B_i$ are disjoint
	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sum_i \Pr \left({A \mid B_i}\right)\Pr \left({B_i}\right)\)	\(\displaystyle \)	by definition of conditional probability

$\blacksquare$

Also see

• Bayes' Theorem, which is related.

Alternative Names

This theorem is also called the Partition Theorem, but as there are already quite a few theorems with such a name (with some guy's name appended to it), it can be argued that it is a good idea to use this somewhat more distinctive name. Grimmett and Welsh appear to be dismissive of them.

Other names include:

• Law of Alternatives
• Theorem of Total Probability

Sources

• Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction (1986): $\S 1.8$: Theorem $1 \text{B}$