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8/8/2019 (8.4) the Addition Priniciple of Counting
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LESSON (8.4) The
ADDITION PRINCIPLE
OF COUNTING ANDVENN DIAGRAMS
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Problem Solving with a Venn
Diagram
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Next we will look at Venn Diagrams.
In a Venn Diagram the box represents
the entire sample space.
A B
Members
that fitEvent A
go in this
circle.
Members
that fitEvent B
go in this
circle.
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A B A B
Event A and B Event A or B
Which is ³A and B´?
Which is ³A or B´?
This is calledINTERSECTION.
This is calledUNION.
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A BA BA BA B
+ _
=
The ADDITION PRINCIPLE OF COUNTING
P(A or B) = P(A) + P(B) - P(A and B)
A B
But we haveadded this piece twice! That isone extra time!
We need tosubtract off the extra
time!
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Example #1)
Given the following probabilities:
P(A)=0.8 P(B)=0.3 P(A and B)=0.2
Find the P(A or B).
This can be solved two ways.
1. Using Venn Diagrams2. Using the formula
We will solve it both ways.
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Example #1 (continued)
P(A)=0.8 P(B)=0.3 P(A and B)=0.2
Find the P(A or B).
Solution using Venn Diagrams:
A B In this example wewill fill up theVenn Diagram
with probabilities.
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Solution using Venn Diagrams:
A B
First fill inwhere the events
overlap.
The probability that a student fits the event A and B
is 0.2.
That means theentire A circle
must add up to0.8.
0.20.6 0.1
0.1
The probability that a student fits the event B is 0.3.The box represents the entire sample
space and mustadd up to 1.
0.2
0.1
0.10.6
The probability that a student fits the event A is 0.8.
That means theentire B circle
must add up to0.3.
Example #1 (continued)
P(A)=0.8 P(B)=0.3 P(A and B)=0.2
Find the P(A or B).
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Then find the probability of A or B.
A B
0.20.6 0.10.2
.1
0.10.6
P(A or B) = 0.6 + 0.2 + 0.1
I will start byshading A or B.
Then I will add up the probabilities in
the shaded area.
= 0.9 Answer
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Solution using the formula:
P(A or B) = P(A) + P(B) - P(A and B)
= 0.8 + 0.3 - 0.2
= 0.9
Example #1 (continued)
P(A)=0.8 P(B)=0.3 P(A and B)=0.2
Find the P(A or B).
Answer
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Example #2.)
There are 50 students. 18 are takingEnglish. 23 are taking Math. 10 are
taking English and Math.
If one is selected at random, find theprobability that the student is taking
English or Math.
E = taking English
M = taking Math
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Solution using Venn Diagrams:
E MIn this example
we will fill up theVenn Diagram
with the number of students.
Example #2 (continued) There are 50 students.
18 are taking English. 23 are taking Math. 10
are taking English and Math.
If one is selected at random, find the probabilitythat the student is taking English or Math.
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Solution using Venn Diagrams:
E M
First fill inwhere the events
overlap.
The number of students taking
English and Math
is 10.
That means thenumber of students taking
English must add
up to 18.
108 13
19
The number of students takingMath is 23.
The box represents the entire sample
space and mustadd up to 50.
10
19
138
The number of students takingEnglish is 18.
That means thenumber of
students takingMath must add up
to 23.
Example #2 (continued) There are 50 students.
18 are taking English. 23 are taking Math. 10
are taking English and Math.
If one is selected at random, find the probabilitythat the student is taking English or Math.
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Then find the probability of English or Math.
E M
108 1310
19
138
P(E or M) =
I will start byshading E or M.
ThenI
will find the probability in theshaded area.
= 0.62
8 10 13
50
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Solution using the formula:
P(E or M) = P(E) + P(M) - P(E and M)
= 0.62
Example #2 (continued) There are 50 students. 18
are taking English. 23 are taking Math. 10 are
taking English and Math.If one is selected at random, find the probability
that the student is taking English or Math.
18 23 10
50 50 50
!
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In a class of 30 students, 21 belong to a
sports team,16 belong to the band and4 belong to neither. How many students
belong to both the band and a sports team?
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(16)
(30)
(4)(21)
Neither
Both
Whole Class
Sports Band
And that 16 ± 11 = 5 students belong ONLY to the band.This means that 21 ± 11 = 10 students belong ONLY to a sports team.
1110 5
In a class of 30
students, 21
belong to a
sports team,16
belong to the
band and
4 belong to
neither. How
many students
belong to both
the band and a
sports team?
How many students are taking sports and/or band?
30 ± 4 = 26 in the combined circles
How many students in sports AND band?
37 ± 26 = 11 in Both