Joint probability describes the probability of two or more events happening at the same time. It is denoted by $P(A \cup B)$ for the event that $A$ or $B$ occurs and $P(A \cap B)$ for the event that $A$ and $B$ occurs where $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ $P(A \cap B) = P(A) + P(B) - P(A \cup B)$ ## Union of many events To calculate the union of many events, we need to adjust for overcounting. For example, the union of $A$ and $B$ can be calculated as $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. We subtract the intersection because it is included once in $P(A)$ and again in $P(B)$, so it is double counted. To extend this idea, we can use the general formula $P(\text{Union of many events}) = P(\text{Singles}) - P(\text{Doubles}) + P(\text{Triples}) - P(\text{Quadruples}) \dots$ where the joint probabilities of even numbered events are subtracted and the odd numbered events are added back again. ## Union of disjoint events When all events are disjoint, there will be no overlap and so the probability of event A or event B occurring is simply $P(A \cup B) = P(A) + P(B)$ The probability of the intersection of A and B is always 0 for disjoint events. [[chain rule of probabilities]]