Total Probability Theorem

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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:

$\displaystyle \forall A \in \Sigma: \map \Pr A = \sum_i \map \Pr {A \mid B_i} \, \map \Pr {B_i}$


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 $B_i$.

Then:

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


Proof

\(\displaystyle \map \Pr A\) \(=\) \(\displaystyle \map \Pr {A \cap \paren {\bigcup_i B_i} }\) Intersection with Subset is Subset: $\bigcup_i B_i = \Omega$ and $A \subseteq \Omega$
\(\displaystyle \) \(=\) \(\displaystyle \map \Pr {\bigcup_i \paren {A \cap B_i} }\) Intersection Distributes over Union
\(\displaystyle \) \(=\) \(\displaystyle \sum_i \map \Pr {A \cap B_i}\) as all the $A \cap B_i$ are disjoint
\(\displaystyle \) \(=\) \(\displaystyle \sum_i \map \Pr {A \mid B_i} \map \Pr {B_i}\) 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


Sources