Adding and Multiplying Probabilities

If we have two events, A and B, the probability of A or B is the probability of A added to the probability of B less the probability of both A and B, or:

P(A or B) = P(A) + P(B) - P(A and B).

This equation is often called the law of addition.

For example, what is the probability of drawing a card from a standard deck of cards and getting a face card or a club? The probability of drawing a club is 13/52. The probability of drawing a face card is 12/52. However, three of the clubs are also face cards, so there are only 22 of the 52 cards that are either clubs or face cards. If we simply add 13/52 and 12/52, we count three cards twice. We should only count them once, and hence we subtract out the 3/52.

Our answer is then 13/52 + 12/52 - 3/52.

What is the probability of getting both A and B? A formula for that is called the law of multiplication:

P(A and B) = P(A)*P(B|A) = P(B)*P(A|B)

The P(A|B) is called a conditional probability and is read as "the probability of A given B."

What is the probability of getting a face card and a club? The formula says that we need two things, the probability of getting a face card (12/52) and the probability of getting a club given that we have a face card (1/4). If we put this into the formula, we get:

P(face card and club) = (12/52)*(1/4)=3/52.

If we do it as P(club and face card) = (13/52)*(3/13)=3/52, we get the same answer.

If P(A and B) = 0 then A and B are mutually exclusive events, meaning that if one happens, the other cannot.

If P(A|B)=P(A) then A and B are independent events, that is, the probability of getting an A does not depend on B. There are many statistical results that depend on independence.