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Joint Probability

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(AB)=P(A,B)=P(AB)P(B)P(A \cap B) =P(A,B) = P(A|B)*P(B)
Anyway, if an event A and B are idenpendent then
P(AB)=P(A)P(A|B) = P(A)
, and therefore,
P(AB)=P(A,B)P(A \cap B) = P(A,B)
become
P(A)P(B)P(A) * P(B)
Example: The probability that a card is a five and black = p(five and black)
Here p(five|black) is the conditional probability of drawing a five given that the card is black, and p(black) is the probability of drawing a black card.
The probability of drawing a black card from a standard deck of 52 playing cards is:
p(black) = Number of black cards / Total number of cards
= 26 / 52
= 1/2
The probability of drawing a five given that the card is black is:
p(five|black) = Number of black fives / Number of black cards
= 2 / 26
= 1/13
Therefore, the joint probability of drawing a five and a black card is:
p(five and black) = p(five|black) * p(black)
= (1/13) * (1/2)
= 1/26
Another way
2/52 = 1/26 (There are two black fives in a deck of 52 cards, the five of spades and the five of clubs).

OR Events

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