# Probability¶

## Overview¶

### Probability Space¶

A probability space: $$(\Omega, \mathcal{F}, P)$$ where

• $$\Omega$$ is the sample space (set of all possible outcomes)

• $$\sigma$$-algebra $$\mathcal{F}$$ is a set of events where each event is an outcome

• $$P$$ is a function giving the probability of an event

#### Probability Function ($$P$$)¶

Probability of an event must be greater than or equal to zero, probability of the sample space is equal to one, and probabilities of an intersection are additive:

• $$P(A) \geq 0$$

• $$P(\Omega) = 1$$

• $$P(A_1 \cup A_1 \cup ...) = P(A_1) + P(A_2) + ...$$ for any countable collection of mutually exclusive events $$A_1, A_2, ...$$

#### Consequences¶

• $$P(o_i) = 1/n$$ in a sample space $$\Omega = \{ o_1, o_2, ..., o_n\}$$ where each outcome $$o_i$$ is equally likely to occur $$\hspace{5pt} \forall \hspace{5pt} i = 1, ..., n$$

• $$P(\bar{A}) = 1 - P(A)$$ where $$\bar{A}$$ is the complement of event $$A$$

• For events $$A$$ and $$B$$, $$P(A\cup B) = P(A) + P(B) - P(A\cap B)$$

• If $$A\subseteq B$$, then $$P(A) \leq P(B)$$

• For any event A, $$P(A) = \sum_{k=1}^m P(A \cap B_k)$$ where mutually exclusive events $$B_1, B_2, ..., B_m$$ such that $$B_1 \cup B_2 \cup ... \cup B_m$$ and $$P(B_i) > 0 \hspace{5pt} \forall \hspace{5pt} i$$

## Sources¶

Contributions made by our wonderful GitHub Contributors: @wyattowalsh