Conditional Probability

Compact study note.

Summary

Conditional probability updates the probability of an event after restricting attention to another event that occurred. It is the basis for Bayes' theorem and conditional distributions.^[1]

Prerequisites

• Probability Space

Notation and Assumptions

For events $A,B\in \mathcal{F}$ with $P(B)>0$ :

Essential Result

Conditioning changes the denominator from the whole sample space to the event being conditioned on.

Small Example

In a fair die roll, condition on result being at least $4$ . The even outcomes inside that condition are $4$ and $6$ , so

Common Mistakes

• Conditioning on an event with probability zero without additional theory.
• Assuming reversed conditional probabilities are equal.

Connections

• Total Probability and Bayes' Theorem
• Independence
• Joint Marginal and Conditional Distributions

References