Download Section 11.2

Chapter 11
Probability
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Chapter 11: Probability
11.1
11.2
11.3
11.4
11.5
Basic Concepts
Events Involving “Not” and “Or”
Conditional Probability and Events
Involving “And”
Binomial Probability
Expected Value and Simulation
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Section 11-2
Events Involving “Not” and “Or”
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Events Involving “Not” and “Or”
• Know that the probability of an event is a real
number between 0 and 1, inclusive of both, and
know the meanings of the terms impossible
event and certain event.
• Understand the correspondences among set
theory, logic, and arithmetic.
• Determine the probability of “not E” given the
probability of E.
• Determine the probability of “A or B” given the
probabilities of A, B, and A and B.
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Properties of Probability
Let E be an event from the sample space S. That is, E
is a subset of S. Then the following properties hold.
1. 0  P( E )  1
(The probability of an event is
between 0 and 1, inclusive.)
2. P ()  0
(The probability of an impossible
event is 0.)
3. P ( S )  1
(The probability of a certain event
is 1.)
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Example: Finding Probability When
Rolling a Die
When a single fair die is rolled, find the
probability of each event.
a) the number 3 is rolled
b) a number other than 3 is rolled
c) the number 7 is rolled
d) a number less than 7 is rolled
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Example: Finding Probability When
Rolling a Die
Solution
There are six possible outcomes for the die:
{1, 2, 3, 4, 5, 6}.
1
a) the number 3 is rolled P (3) 
6
5
P(not 3) 
6
b) a number other than 3 is rolled
c) the number 7 is rolled
P (7)  0
d) a number less than 7 is rolled
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P(less than 7)  1
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Events Involving “Not”
The table on the next slide shows the
correspondences that are the basis for the
probability rules developed in this section. For
example, the probability of an event not happening
involves the complement and subtraction.
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Correspondences
Set Theory
Logic
Arithmetic
Operation or
Connective
(Symbol)
Complement
Not
Subtraction
( )
( )
()
Operation or
Connective
(Symbol)
Union
Or
Addition
( )
()
()
Operation or
Connective
(Symbol)
Intersection
And
Multiplication
( )
()
()
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Probability of a Complement
The probability that an event E will not occur is
equal to one minus the probability that it will occur.
E
S
E
P(not E )  P( S )  P( E )
 1  P( E )
So we have
P( E )  P  E    1
and P( E )  1  P( E).
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Example: Finding the Probability from
a Complement
When a single card is drawn from a standard 52-card
deck, what is the probability that it will not be an ace?
Solution
P(not an ace)  1  P(ace)
4
1
52
48 12

 .
52 13
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Events Involving “Or”
Probability of one event or another should
involve the union and addition.
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Mutually Exclusive Events
Two events A and B are mutually exclusive events
if they have no outcomes in common. (Mutually
exclusive events cannot occur simultaneously.)
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Addition Rule of Probability (for A or B)
If A and B are any two events, then
P( A or B)  P( A)  P( B)  P( A and B).
If A and B are mutually exclusive, then
P( A or B)  P( A)  P( B).
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Example: Finding the Probability of an
Event Involving “Or”
When a single card is drawn from a standard 52-card
deck, what is the probability that it will be a king or a
diamond?
Solution
P(king or diamond)  P(K)  P(D)  P(K and D)
4
13
1



52
52
52
16 4


52 13
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Example: Finding the Probability of an
Event Involving “Or”
If a single die is rolled, what is the probability of a
2 or odd?
Solution
These are mutually exclusive events.
P(2 or odd)  P(2)  P(odd)
1
3
4 2


  .
6
6
6 3
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Example: Finding the Probability of an
Event Involving “Or”
Of 20 elective courses, Emily plans to enroll in one,
which she will choose by throwing a dart at the
schedule of courses. If 8 of the courses are
recreational, 9 are interesting, and 3 are both
recreational and interesting, find the probability that
the course she chooses will have at least one of
these two attributes.
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Example: Finding the Probability of an
Event Involving “Or”
Solution
If R denotes “recreational” and I denotes
“interesting,” then
8
9
3
P( R)  , P( I )  , P( R and I) =
20
20
20
R and I are not mutually exclusive.
8
9
3
P( R or I ) 


20 20 20
14 7


20 10
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