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:

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

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)

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

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