General Addition Rule AP Statistics

Addition Rule for Disjoint Events If two events A and B are disjoint: P(A or B) = P(A) + P(B) If three events are disjoint: P(A or B or C) = P(A) + P(B) + P(C)

Mutually Exclusive (Disjoint) Events Two events have no outcomes in common. Example: An animal can’t be a dog and a cat at the same time. Example: Roll a “2” or a “5” – these can’t happen at the same time. Note: the second “or” statement involves only one roll of the die, not two rolls.

Venn Diagram - Disjoint Events B P(A or B) = P (A) + P(B) Can also be written P(A B). This is the union of events A and B.

Non –Disjoint Events: Can Have Outcomes in Common. If two events E & F are not disjoint, P(E or F) = P(E) + P(F) – P(E and F) This is The General Addition Rule. P(E and F) is called a joint probability and can be written P(E∩F). P(E∩F) is also called the intersection of events E and F.

Non Disjoint Events A B The shaded region shows the intersection of events A and B, where A and B can happen at the same time.

Example Event A = (Being a senior at CRHS) Event B = (Taking Statistics) You can have seniors at CRHS who are taking Stats! These events can happen at the same time.

Example 6.17 on page 362 P (Deb becoming a partner) = .7 P (Matt becoming a partner) = .5 P (Deb and Matt becoming a partner) = .3 P(Deb or Matt becoming a partner) = 1.2?? This probability exceeds 1 which is impossible!

Venn Diagram Interpretation D and M D and MC DC and M DC and MC Let’s add in all the values for these 4 joint probabilities!

Example 6.17 Continued P(at least one is promoted) = P(Deb or Matt) = .7 + .5 - .3 = .9 P(neither is promoted) = 1-.9 = .1 The reason that we “correct” by subtracting .3 (the intersection of Deb and Matt) is that if we don’t, it is counted twice.

Union The union of event A with event B consists of all outcomes that are in at least one of the two events. Example: Event E is rolling a die and getting prime number or even number. E = {2,3,4,5,6}

Venn Diagram - A or B A B Event A consists of {2,3,5}. Event B consists of {2,4,6}. A U B consists of {2,3,4,5,6}

Intersection This is the event where both A and B happen. It consists of all outcomes that are in both events. Denoted:

Venn Diagram - A and B A B From the previous example, A∩B is {2}, which is both an even and prime number.

General Rule for the Union of Two Events P(A or B) = P(A) + P(B) – P(A and B). Note: if there is no intersection (if A and B are mutually exclusive), then the term: P(A and B) is equal to zero, which returns us to the Addition Rule for Disjoint Events.

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