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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
A
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}
E  AB
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: E  A  B
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.
Homework
• Worksheet