Two-Way Tables• When finding probabilities involving two events, a

two-way table can make the calculations easier.

College statistics students wanted to find out how common it is for young adults to have their ears pierced. They recorded data on two variables – gender and whether the student had pierced ears, for all 178 people in the class.

If we choose a student at random,What is the probability thata. They have pierced earsb. They are male with pierced earsc. They are male or have pierced ears

Pierced EarsGender Yes No TOTAL

Male 19 71 90

Female 84 4 88

TOTAL 103 75 178

Venn-Diagrams• If events A and B are not mutually exclusive,

they can occur together• The probability that one or the other occurs is

less than the sum of their probabilities

Venn-Diagrams• A Venn diagram takes of the double counting

problem using the General Addition Rule of Two Events:

P(A or B) = P(A) + P(B) – P (A and B)

If two events are mutually exclusive, the P(A and B) = 0 - This is just a special case of the addition rule since P(A and B) = 0, we are subtracting nothing

P(A or B) = P(A) + P(B)

Venn-DiagramsVocabulary and Standard Notation:• The complement of an event AC contains

the outcomes that are not in A

• Events A and B are mutually exclusive (disjoint) if they do not overlap, that have no outcomes in common.

Venn-DiagramsVocabulary and Standard Notation:• The event A and B is the intersection

of A and B, and it is notated as A ∩ B

• The event A or B is the union of A and B, and it is notated as A U B

Venn-DiagramsThe notation that will be used on the AP Exam that is also on the equation sheet:

P(A U B) = P(A) + P(B) – P(A ∩ B) or and

• A U B represents union (or)• A ∩ B represents intersection (and)

Pg. 306-307