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

Theorem for Conditional Probabilities
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 $\forall i: \Pr \left({B_i}\right) > 0$.

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

Then:
 * $\displaystyle \forall A \in \Sigma: \Pr \left({A \mid C}\right) = \sum_i \Pr \left({A \mid C \cap B_i}\right) \Pr \left({B_i}\right)$

Proof
First define $Q_C := \Pr \left({\, \cdot \mid C}\right)$.

Then, from Conditional Probability Defines Probability Space, $\left({\Omega, \Sigma, Q_C}\right)$ is a probability space.

Therefore the Total Probability Theorem also holds true.

Hence we have:

Also known as
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. and appear to be dismissive of them.

Other names include:
 * Law of Alternatives
 * Law of Total Probability
 * Theorem of Total Probability

Also see

 * Bayes' Theorem