Dice.jpgThere are several rules that can help solve probability problems and these rules can often be illustrated using Venn diagrams.

Complementary Events

If is an event then not A is called the complementary event.

On a Venn diagram:


P(A) + P(A') = 1


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 Y12_Probability_Rules_03.gif B) = P(A) + P(B)

This is sometimes known as the Addition Law for probabilities.

Note that for mutually exclusive events P(A Y12_Probability_Rules_04.gif B) = 0


When a die is thrown P( even number OR a five ) = Y12_Probability_Rules_05.gif

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 Y12_Probability_Rules_03.gif B) = P(A) + P(B) − P(A Y12_Probability_Rules_04.gif B)

It is essential to subtract the probability of A and B occurring together or this probability will be counted twice!


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)

                                  = Y12_Probability_Rules_07.gif

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 .


When a coin is tossed and a die is thrown, P(head AND an even number) = 0.5 x 0.5 = 0.25


Probability Rules
Key words
Operation with probabilities
Mutually Exclusive events
Independent events