# Definition:Conditional Probability

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## Definition

Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$.

Let $A, B \in \Sigma$ be events of $\EE$.

We write the **conditional probability of $A$ given $B$** as $\map \Pr {A \mid B}$, and define it as:

*the probability that $A$ has occurred, given that $B$ has occurred.*

## Also see

- Chain Rule for Probability, where it is shown that $\map \Pr {A \mid B} = \dfrac {\map \Pr {A \cap B} } {\map \Pr B}$.

- Results about
**conditional probabilities**can be found here.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**conditional probability**

- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 1.6$: Conditional probabilities: $(19)$ - 1988: Dominic Welsh:
*Codes and Cryptography*... (previous) ... (next): Notation