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(BA) = P(B)*P(AB)
The P(AB) 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(AB)=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.
