Total Probability Theorem

Theorem

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

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

Then:

$\ds \forall A \in \Sigma: \map \Pr A = \sum_i \condprob A {B_i} \map \Pr {B_i}$

Theorem for Conditional Probabilities

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

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

Let $C \in \Sigma$ be an event independent to any of the $B_i$.

Let $\map \Pr C > 0$.

Then:

$\ds \forall A \in \Sigma: \condprob A C = \sum_i \condprob A {C \cap B_i} \, \map \Pr {B_i}$

Proof

\(\ds \map \Pr A\)

\(=\)

\(\ds \map \Pr {A \cap \paren {\bigcup_i B_i} }\)

Intersection with Subset is Subset: $\ds \bigcup_i B_i = \Omega$ and $A \subseteq \Omega$

\(\ds \)

\(=\)

\(\ds \map \Pr {\bigcup_i \paren {A \cap B_i} }\)

Intersection Distributes over Union

\(\ds \)

\(=\)

\(\ds \sum_i \map \Pr {A \cap B_i}\)

as all the $A \cap B_i$ are disjoint

\(\ds \)

\(=\)

\(\ds \sum_i \condprob A {B_i} \map \Pr {B_i}\)

Definition of Conditional Probability

$\blacksquare$

Also known as

This theorem is also called the Partition Theorem, but as there are already quite a few theorems with such a name, $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers to use this somewhat more distinctive name.

Other names include:

Law of Alternatives

Law of Total Probability

Total Probability Law

Theorem of Total Probability

Grimmett and Welsh appear to be dismissive of them:

This theorem has several other fancy names, such as 'the theorem of total probability'; ... from Probability: An Introduction

Also see

• Bayes' Theorem

Sources

• 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $\S 1.8$: The partition theorem: Theorem $1 \text{B}$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): total probability law