Chapter 11
Sequences,
Induction, and
Probability
11.7 Probability
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Objectives:
•
•
•
•
•
Compute empirical probability.
Compute theoretical probability.
Find the probability that an event will not occur.
Find the probability of one event or a second event
occurring.
Find the probability of one event and a second event
occurring.
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Probability
Probabilities of events are expressed as numbers
ranging from 0 to 1, or 0% to 100%. The closer the
probability of a given event is to 1, the more likely it is
that the event will occur. The closer the probability of a
given event is to 0, the less likely it is that the event will
occur.
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Empirical Probability
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Example: Empirical Probabilities with Real-World Data
The data in the table are based on 100,000 U.S. women,
ages 40 to 50, who participated in mammography
screening. Find the probability that a woman aged 40 to 50
has a positive mammogram.
720  6944
7664
479


 0.077
100,000
100,000 6250
The probability
is 77%.
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Example: Empirical Probabilities with Real-World Data
The data in the table are based on 100,000 U.S. women,
ages 40 to 50, who participated in mammography
screening. Among women with breast cancer, find the
probability of a positive mammogram.
720
720
9

  0.9
720  80 800 10
The probability
is 90%.
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Example: Empirical Probabilities with Real-World Data
The data in the table are based on 100,000 U.S. women,
ages 40 to 50, who participated in mammography
screening. Among women with positive mammograms,
find the probability of having breast cancer.
720
720
45


 0.094
720  6944 7664 479
The probability
is 9.4%.
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Theoretical Probability
Any occurrence for which the outcome is uncertain is
called an experiment. The set of all possible outcomes
of an experiment is the sample space of the experiment,
denoted by S. An event, denoted by E, is any
subcollection, or subset, of a sample space. Theoretical
probability applies to situations in which the sample
space only contains equally likely outcomes, all of
which are known.
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Computing Theoretical Probability
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Example: Computing Theoretical Probability
A die is rolled. Find the probability of getting a number
greater than 4.
The sample space of equally likely outcomes is
S = {1, 2, 3, 4, 5, 6}.
The event of getting a number greater than 4 can be
n( E )  2
represented by E = {5, 6}.
The probability of getting a number greater than 4 is
n( E ) 2 1
P( E ) 
  .
n( S )
6 3
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The Probability of an Event Not Occurring
The probability that an event E will not occur is equal to
1 minus the probability that it will occur.
P(not E) = 1 – P(E)
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Example: The Probability of an Event Not Occurring
If one person is randomly selected from the world
population represented by the figure, find the probability
that the person does not live in North America. Express
the probability as a simplified fraction and as a decimal
rounded to the nearest thousandth.
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Example: The Probability of an Event Not Occurring
(continued)
P(does not live in North America)
= 1 – P(lives in North America)
550
7000 550
 1


7000 7000 7000
6450 129

 0.921

7000 140
The probability that a person does
129
not live in North America is
or 9.21%.
140
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Or Probabilities with Mutually Exclusive Events
If it is impossible for any two events, A and B, to occur
simultaneously, they are said to be mutually exclusive.
If A and B are mutually exclusive events, then
P(A or B) = P(A) + P(B).
Using set notation, P( A B)  P( A)  P( B).
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Example: The Probability of Either of Two Mutually
Exclusive Events Occurring
If you roll a single, six-sided die, what is the probability
of getting either a 4 or a 5?
P(getting either a 4 or a 5) = P(4) + P(5)
1 1 2 1
   
6 6 6 3
1
The probability of getting either a 4 or a 5 is .
3
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Or Probabilities with Events That Are Not Mutually
Exclusive
If A and B are not mutually exclusive events, then
P(A or B) = P(A) + P(B) – P(A and B).
Using set notation, P( A B)  P( A)  P( B)  P( A B).
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Example: An Or Probability with Real-World Data
If one person is randomly selected from the population
represented in the table, find the probability that the person
is married or female.
P(married or female)
= P(married) + P(female) – P(married and female)
130 124 65 189




 0.78
242 242 242 242
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Example: An Or Probability with Real-World Data
If one person is randomly selected from the population
represented in the table, find the probability that the person
is divorced or widowed.
P(divorced or widowed)
= P(divorced) + P(widowed) – P(divorced and widowed)
24 14
0
38 19





 0.16
242 242 242 242 121
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And Probabilities with Independent Events
Two events are independent events if the occurrence of
either of them has no effect on the probability of the
other.
If A and B are independent events, then
P(A and B)  P( A) P( B).
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Example: And Probability with Independent Events
Find the probability of a family having four boys in a row.
1111 1
P(four boys in a row) 

2 2 2 2 16
1
The probability of a family having four boys in a row is .
16
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