Addition Rule for Probability

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Addition RuleforProbability

• Vicki Borlaug
• Walters State Community College
• Morristown, Tennessee
• Spring 2006

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This is Rita.
Are the statements TRUE or FALSE?
Rita is playing the violin and soccer.
Rita is playing the violin or soccer.
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Elm or Maple
Elm and Maple
This is called UNION.
This is called INTERSECTION.
Which one is Elm and Maple?
Which one is Elm or Maple?
Like when you put the North and the South
together.
Like when two streets cross.
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Next we will look at Venn Diagrams.
In a Venn Diagram the box represents the entire
sample space.
Members that fit Event A go in this circle.
Members that fit Event B go in this circle.
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Event A and B
Event A or B
This is called INTERSECTION.
This is called UNION.
Which is A and B?
Which is A or B?
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The Addition Rule for Probability
_


P(A or B)
P(A)
P(B)
- P(A and B)
But we have added this piece twice! That is one
extra time!
We need to subtract 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 Diagrams
2. 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
In this example we will fill up the Venn Diagram
with probabilities.
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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
The probability that a student fits the event B
is 0.3.
The probability that a student fits the event A
is 0.8.
First fill in where the events overlap.
The box represents the entire sample space and
must add up to 1.
That means the entire A circle must add up to 0.8.
The probability that a student fits the event A
and B is 0.2.
That means the entire B circle must add up to
0.3.
0.6
0.6
0.2
0.1
0.2
0.1
0.1
0.1
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Then find the probability of A or B.
I will start by shading A or B.
0.6
0.6
0.2
0.1
0.2
0.1
Then I will add up the probabilities in the
shaded area.
0.1
P(A or B)
0.6 0.2 0.1
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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 the formula
P(A or B) P(A) P(B) - P(A and B)
0.8 0.3 - 0.2
0.9
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Example 2.)
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.
E taking English M taking Math
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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.
Solution using Venn Diagrams
In this example we will fill up the Venn
Diagram with the number of students.
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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.
Solution using Venn Diagrams
The number of students taking English is 18.
The number of students taking Math is 23.
First fill in where the events overlap.
The box represents the entire sample space and
must add up to 50.
That means the number of students taking Math
must add up to 23.
That means the number of students taking English
must add up to 18.
The number of students taking English and Math is
10.
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8
10
13
10
13
19
19
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Then find the probability of English or Math.
I will start by shading E or M.
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8
10
13
10
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Then I will find the probability in the shaded
area.
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P(E or M)
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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.
Solution using the formula
P(E or M) P(E) P(M) - P(E and M)
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Class Activity 1)
There are 1580 people in an amusement park. 570
of these people ride the rollercoaster. 700 of
these people ride the merry-go-round. 220 of
these people ride the roller coaster and
merry-go-round.
If one person is selected at random, find the
probability that that person rides the roller
coaster or the merry-go-round. a.) Solve using
Venn Diagrams. b.) Solve using the formula for
the Addition Rule for Probability.
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Example 3) Population of apples and pears.
Each member of this population can be described
in two ways.
1. Type of fruit
2. Whether it has a worm or not
We will make a table to organize this data.
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Example 3) Population of apples and pears.
no worm
worm
8
?
5
3
?
apple
?
4
6
2
?
pear
5
9
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Ex. 3 (continued)
Experiment One is selected at random.
Find the probability that . . .
a.) . . . it is a pear and has a worm.
b.) . . . it is a pear or has a worm.
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Ex. 3 (continued)
Solution to 3a.)
P(pear and worm)
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Ex. 3 (continued)
Solution to 3b.)
P(pear or worm)
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Ex. 3 (continued)
Alternate Solution to 3b.)
P(pear or worm)
P(pear) P(worm) P (pear and worm)
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Class Activity 2)
There are our modes of transportation horse,
bike, canoe. Each has a person or does not
have a person.
1.) Make a table to represent this data.
2.) If one is selected at random find the
following
a.) P( horse or has a person)
b.) P( horse and has a person)
c.) P( bike or does not have a person)
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The end!