Definition: Law of conditional probability

Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Probability ▷ Conditional probability

Definition: (law of conditional probability, also called “product rule”) Let $A$ and $B$ be two arbitrary statements about random variables, such as statements about the presence or absence of an event or about the value of a scalar, vector or matrix. Furthermore, assume a joint probability distribution $p(A,B)$. Then, $p(A \vert B)$ is called the conditional probability that $A$ is true, given that $B$ is true, and is given by

$$\label{eq:prob-cond} p(A|B) = \frac{p(A,B)}{p(B)}$$

where $p(B)$ is the marginal probability of $B$.

 

Sources:

• Wikipedia (2020): "Conditional probability"; in: Wikipedia, the free encyclopedia, retrieved on 2020-05-10; URL: https://en.wikipedia.org/wiki/Conditional_probability#Definition.
• Jason Browlee (2019): "A Gentle Introduction to Joint, Marginal, and Conditional Probability"; in: Machine Learning Mastery, retrieved on 2021-08-01; URL: https://machinelearningmastery.com/joint-marginal-and-conditional-probability-for-machine-learning/.

Metadata: ID: D51 | shortcut: prob-cond | author: JoramSoch | date: 2020-05-10, 20:06.