- [[joint probability of independent events]], [[derive Bayes rule]], [[conditional probability]] # Idea Joint probability is denoted as $P(A \ \text{and} \ B)$ or $P(A \cap B)$ or $P(A, B)$ (read "the probability of A and B jointly occurring"). It's also known as the intersection of $A$ and $B$. ![[s20220326_012031.png|300]] The joint probability of **dependent** events (i.e., events that influence each other) is given by $P(A,B) = P(A|B) \ P(B)$ $P(B,A) = P(B|A) \ P(A)$ Note that the two equations above are identical. Contrast with [[joint probability of independent events]], which is just a special case of the equations above where $A$ and $B$ are not conditional on each other. That is, the probability of $A$ and $B$ jointly occurring, $P(A,B)$, is the probability of $A$ occurring, $P(A)$, multiplied by the probability of $B$ occurring given that $A$ has already occurred, $P(B|A)$ (that is, $B$ is [[conditional probability|conditional]] on $A$). The [[conditional probability]], $P(B|A)$ is used to calculate joint probability. Note that the equations above are used to [[derive Bayes rule]] (see [[Bayes theorem]]). **Dependent** events: If event $A$ is drawing a king from deck of 52 cards and event $B$ is drawing another a second king from the deck (without replacing the king after event $A$), then the probabilities are dependent: $P(A,B) = \frac{4}{52} \cdot \frac{3}{51}$ **Independent** events (see [[joint probability of independent events]]): If I replace the king after drawing it, then the events are independent. $P(A,B) = \frac{4}{52} \cdot \frac{4}{52}$ # References - [Joint Probability Definition](https://www.investopedia.com/terms/j/jointprobability.asp)