# 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 $\condprob A B$, 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 $\condprob A B = \dfrac {\map \Pr {A \cap B} } {\map \Pr B}$.

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

## Technical Note

The $\LaTeX$ code for \(\condprob {A} {B}\) is `\condprob {A} {B}`

.

When the arguments are single characters, it is usual to omit the braces:

`\condprob n p`

## Sources

- 1988: Dominic Welsh:
*Codes and Cryptography*... (previous) ... (next): Notation - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**conditional probability** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**conditional probability** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**conditional probability**

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- 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $1$: Events and probabilities: $1.6$: Conditional probabilities: $(19)$