# Total Probability Theorem

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## Contents

## 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

- 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): Entry:**total probability law**