There are several rules that can help solve probability problems and these rules can often be illustrated using Venn diagrams.
Complementary Events
If A is an event then not A is called the complementary event.
On a Venn diagram:
P(A) + P(A') = 1
Example
If the probability that a member of a class has brown eyes is 0.2, the probability that a person does not have brown eyes is 1 − 0.2 = 0.8. ( "having brown eyes" and "not having brown eyes" are complementary events.
Mutually Exclusive Events
If an event A can occur OR an event B can occur but not both A and B can occur then the two events are said to be mutually exclusive.
On a Venn diagram:
P(A OR B) = P(A B) = P(A) + P(B)
This is sometimes known as the Addition Law for probabilities.
Note that for mutually exclusive events P(A B) = 0
Example
When a die is thrown P( even number OR a five ) =
Not Mutually Exclusive Events
If an event A and an event B can occur simultaneously they are not mutually exclusive.
On a Venn diagram:
P(A OR B) = P(A B) = P(A) + P(B) − P(A B)
It is essential to subtract the probability of A and B occurring together or this probability will be counted twice!
Example
When a card is drawn from a pack of 52 playing cards,
P(an ace OR a heart) = P(an ace) + P(a heart) − P(ace of hearts)
=
Independent Events
If either of events A and B can occur without being influenced by one another then the two events are independent.
For independent events P(A AND B) = P(A B) = P(A) x P(B)
This is sometimes known as the Multiplication Law for probabilities .
Example
When a coin is tossed and a die is thrown, P(head AND an even number) = 0.5 x 0.5 = 0.25
Summary
Probability Rules 
Key words

Operation with probabilities

Mutually Exclusive events 
OR

ADD

Independent events 
AND

MULTIPLY
