Definition:Conditional Probability

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

• 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $1$: Events and probabilities: $1.6$: Conditional probabilities: $(19)$
• 1988: Dominic Welsh: Codes and Cryptography ... (previous) ... (next): Notation
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): conditional probability
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): probability
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): conditional probability
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): probability
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): conditional probability