12.3 Mutually Exclusive
Events and
Venn Diagrams
Content Objective: SWBAT correctly
use Venn diagrams as a tool to break
non-mutually exclusive events into
mutually exclusive events in order to
find the probability of these events.
Language Objective: SWBAT
correctly draw Venn diagrams and
write the probability of the events
within.
Mutually Exclusive
When two outcomes or events cannot
both happen they are considered to
be mutually exclusive.
Example:
The tree diagram below represents the
possibilities of success at auditions
by Abby, Bonita, and Chih-Lin.
You have seen mutually exclusive events
when you added probabilities of
different paths.
One path, from left to right,
represents the outcome that Abby
and Chih-Lin are successful but
Bonita is not (outcome AC).
Another outcome is success by all
(outcome ABC).
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These two outcomes cannot both take
place, so they are mutually exclusive.
Suppose that each student has a 0.5
probability of success.
The the probability of any single path in
the tree is .5 x .5 x .5 = .125.
So the probability that either AC or ABC
occurs is the sum of the probabilities on
two particular paths, .125 + .125 which
equals .25.
Venn Diagrams
A tool for breaking down nonmutually exclusive events into
mutually exclusive events is
called a Venn Diagram.
Venn Diagrams
Venn Diagrams
A Venn Diagram consists of circles
which overlap or are separate.
This tool was created by an English
mathematician,
John Venn, during the 1860s
while teaching and studying at
Cambridge University as a way to
organize information.
An example
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Example A
Melissa has been keeping a record
of probabilities of events involving:
I. Her guitar string breaking (Event B).
II. A pop quiz in Math (Event Q).
III. Her team losing in gym class (Event L).
Example A
The eight mutually exclusive events and their
probabilities.
Example A
The region labeled .01 represents the
probability of a really bad day. This is the
intersection of ALL three circle, so Melissa’s
string breaks, she has a pop quiz and her
team loses.
Example A
Although the three events are not
mutually exclusive, they can be
broken into eight mutually exclusive
events. These events and their
probabilities are shown in the Venn
diagram on the next slide.
Example A
What is the meaning of the region labeled .01?
Example A
What is the meaning of the region labeled .03?
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Example A
The region labeled .03 represents the
probability the Melissa’s string will break
AND her team will lose, BUT there will not
be a pop quiz in Math.
Example A
You can find the probability of pop quiz by
adding the four areas that are part of the
pop-quiz circle: .02 + .09 + .01 + .04 = .16
Example A
The probability of a pretty good day, P(not B
and not Q and not L is pictured by the
region outside the circles. So it is .57.
Example A
What is the probability of pop quiz P(Q) today?
Example A
Find the probability of a pretty good day, P(not
B and not Q and not L)? This means no
string breaks, no quiz and no loss.
The Addition Rule for Mutually
Exclusive Events
This is the rule to help you:
If n1, n2, n3, and so on represent
mutually exclusive events, then the
probability that any event in this
collection will occur is the sum of the
probabilities of the individual events.
P(n1 or n2 or n3 or …) = P(n1) + P(n2) + P(n3) + …
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What if you don’t know the
probabilities like Melissa knew in the
previous problem?
How can you figure out the
probabilities of mutually exclusive
events when you know the
probabilities of non-mutually
exclusive events?
Example B
Here is one way to discover
how to figure this out.
Example B
Step 1: “A student takes
mathematics” and “A student takes
science” are two events.
Are these events mutually
exclusive? Why or why not?
Of the 100 students in 12th grade
at Mathland High School, 70 are
enrolled in mathematics, 50 are in
science, 30 are in both subjects,
and 10 are in neither subject.
Example B
Step 1: “A student takes
mathematics” and “A student takes
science” are two events.
Are these events mutually
exclusive? No, because there is
overlap where students can take
both classes.
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Example B
Step 2: Complete a Venn diagram, similar to the
one below, that shows the enrollments in math
and science courses.
Of the 100 students in 12th grade at Mathland High
School, 70 are enrolled in mathematics, 50 are in
science, 30 are in both subjects, and 10 are in
neither subject.
Example B
Step 3: Use the numbers of students in your
Venn diagram to calculate probabilities.
A = 40, B = 30, C = 20, D = 10
Example B
Step 4: Explain why the probability that a
randomly chosen students takes
mathematics or science, P(M or S), does
not equal P(M) + P(S).
Example B
Step 2: Complete a Venn diagram, similar to
the one below, that shows the enrollments
in math and science courses.
A = 40, B = 30, C = 20, D = 10
Example B
Step 3: Use the numbers of students in your
Venn diagram to calculate probabilities.
A = .4, B = .3, C = .2, D = .1
Example B
Step 4: Explain why the probability that a randomly
chosen students takes mathematics or science,
P(M or S), does not equal P(M) + P(S).
P(M) = .7 and P(S) = .5
P(M) + P(S) = 1.2 which is not possible
and should equal .9
This counts the overlap portion where
students take both twice.
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Example B
Example B
Step 5: Create a formula for calculating
P(M or S) that includes the expressions
P(M), P(S) and P(M and S).
Step 5: Create a formula for calculating
P(M or S) that includes the expressions
P(M), P(S) and P(M and S)
P(M or S) = P(M) + P(S) - P(M and S)
The General Addition Rule
The General Addition Rule
If n1 and n2 represent event 1 and
event 2, then the probability that at
least one of the events will occur can
be found by adding the probabilities
of the events and subtracting the
probability that both will occur.
That means:
P(n1 or n2) = P(n1) + P(n2) - P(n1 and n2)
Homework:
p. 684-686 (1-6, 9, 10, 12)
Honors Homework:
p. 684-686 (1-6, 9-12)
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