Joint probability is the probability of two events happening together.

The two events are usually designated event A and event B. In probability terminology, it can be written as:

$P(A \cap B) =P(A,B) = P(A|B)*P(B)$

Anyway, if an event A and B are idenpendent then $P(A|B) = P(A)$, and therefore, $P(A \cap B) = P(A,B)$ become $P(A) * P(B)$

**Example:** The probability that a card is a five and black = p(five and black)

2/52 = 1/26 (There are two black fives in a deck of 52 cards, the five of spades and the five of clubs).

$P(A \cup B) = P(A)+P(B)-P(A \cap B)$