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Statistics
Chapter 3: Probability
Where We’ve Been


Making Inferences about a Population
Based on a Sample
Graphical and Numerical Descriptive
Measures for Qualitative and
Quantitative Data
McClave: Statistics, 11th ed. Chapter 3:
Probability
2
Where We’re Going



Probability as a Measure of
Uncertainty
Basic Rules for Finding Probabilities
Probability as a Measure of Reliability
for an Inference
McClave: Statistics, 11th ed. Chapter 3:
Probability
3
3.1: Events, Sample Spaces
and Probability

An experiment is an act or process of
observation that leads to a single
outcome that cannot be predicted with
certainty.
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.1: Events, Sample Spaces
and Probability

A sample point is the most basic
outcome of an experiment.
An Ace
A four
A Head
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.1: Events, Sample Spaces
and Probability

A sample space of an experiment is
the collection of all sample points.

Roll a single die:
S: {1, 2, 3, 4, 5, 6}
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.1: Events, Sample Spaces
and Probability

Sample points and sample spaces are
often represented with Venn diagrams.
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.1: Events, Sample Spaces
and Probability
Probability Rules for Sample Points


All probabilities must
be between 0 and 1.
0  pi  1
n

The probabilities of all
the sample points
must sum to 1.
McClave: Statistics, 11th ed. Chapter 3:
Probability
p
i
1
i 1
8
3.1: Events, Sample Spaces
and Probability
Probability Rules for Sample Points


All probabilities must
be between 0 and 1
0  pi  1
0 indicates an impossible outcome and 1 a certain outcome.
n

The probabilities of all
the sample points
must sum to 1
Something has to happen.
McClave: Statistics, 11th ed. Chapter 3:
Probability
p
i
1
i 1
9
3.1: Events, Sample Spaces
and Probability

An event is a specific collection of sample
points:

Event A: Observe an even number.
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.1: Events, Sample Spaces
and Probability

The probability of an event is the sum of
the probabilities of the sample points in the
sample space for the event.


Event A: Observe an even number.
P(A) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.1: Events, Sample Spaces
and Probability

Calculating Probabilities for Events





Define the experiment.
List the sample points.
Assign probabilities to sample points.
Collect all sample points in the event of
interest.
The sum of the sample point probabilities is
the event probability.
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.1: Events, Sample Spaces
and Probability

Calculating Probabilities for Events
Define the experiment
If the
number
of
sample
points
List the sample points
gets
too
large,
we
need
a
way
to
 Assign probabilities to sample points
keep
track of how they can be
 Collect all sample points in the event of
combined
interest for different events.
 The sum of the sample point probabilities is
the event probability

McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.1: Events, Sample Spaces
and Probability

If a sample of n elements is drawn from a set
of N elements (N ≥ n), the number of different
samples is
N!
N
.
 
 n  n ! N  n)!
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.1: Events, Sample Spaces
and Probability

If a sample of 5 elements is drawn from a set
of 20 elements, the number of different
samples is
20!
20 19 18   3  2 1
 20 


 15,504.
 
 5  5! 20  5)!  5  4  3  2 115 14 13   3  2 1
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.2: Unions and Intersections
Compound Events
Made of two or
more other events
Union
A B
Either A or B,
or both, occur
Intersection
A B
Both A and B
occur
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.2: Unions and Intersections
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.2: Unions and Intersections
A
AB
AC
ABC
B
BC
C
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.3: Complementary Events

The complement of any event A is the
event that A does not occur, AC.
A: {Toss an even number}
AC: {Toss an odd number}
B: {Toss a number ≤ 3}
BC: {Toss a number ≥ 4}
A  B = {1,2,3,4,6}
[A  B]C = {5}
(Neither A nor B occur)
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.3: Complementary Events
P ( A)  P ( A )  1
C
P ( A)  1  P ( A )
C
P ( A )  1  P ( A)
C
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.3: Complementary Events
A: {At least one head on two coin flips}
AC: {No heads}
A : {HH , HT , TH }
AC : {TT }
P( A)  1 4  1 4  1 4  3 4
P( A )  1 4
C
P( A)  1  P( A )  1  1 4  3 4
C
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.4: The Additive Rule and
Mutually Exclusive Events

The probability of the union of
events A and B is the sum of the
probabilities of A and B minus the
probability of the intersection of A
and B:
P( A  B)  P( A)  P( B)  P( A  B)
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.4: The Additive Rule and
Mutually Exclusive Events
At a particular hospital, the
probability of a patient having
surgery (Event A) is .12, of an
obstetric treatment (Event B)
.16, and of both .02.
What is the probability that a patient will
have either treatment?
P(Surgery or Obstetrics)  P( A B)
P( A B)  P( A)  P( B)  P( A B)
P( A B)  .12  .16  .02  .26
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.4: The Additive Rule and
Mutually Exclusive Events

Events A and B are mutually
exclusive if A  B contains no sample
points.
McClave: Statistics, 11th ed. Chapter 3:
Probability
24
3.4: The Additive Rule and
Mutually Exclusive Events
If A and B are mutually exclusive,
P( A  B)  P( A)  P( B)
McClave: Statistics, 11th ed. Chapter 3:
Probability
25
3.5: Conditional Probability

Additional information or other events
occurring may have an impact on the
probability of an event.
McClave: Statistics, 11th ed. Chapter 3:
Probability
26
3.5: Conditional Probability

Additional information may
have an impact on the
probability of an event.


P(Rolling a 6) is one-sixth
(unconditionally).
If we know an even number was
rolled, the probability of a 6 goes
up to one-third.
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.5: Conditional Probability


The sample space is reduced to only the
conditioning event.
To find P(A), once we know B has occurred (i.e.,
given B), we ignore BC (including the A region
within BC).
BC
B
A
A
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.5: Conditional Probability
P( A  B)
P( A B) 
P( B)
BC
B
A
A
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.5: Conditional Probability
P( A  B)
P( A B) 
P( B)
B
A
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.5: Conditional Probability

55% of sampled executives had cheated at
golf (event A).


20% of sampled executives had cheated at
golf and lied in business (event B).


P(A) = .55
P(A  B) = .20
What is the probability that an executive had
lied in business, given s/he had cheated in
golf, P(B|A)?
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.5: Conditional Probability

P(A) = .55

P(A  B) = .20

What is P(B|A)?
P( B  A)
P( B A) 
P( A)
.20
P ( B A) 
 .364
.55
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.6: The Multiplicative Rule
and Independent Events

The conditional
probability formula can
be rearranged into the
Multiplicative Rule of
Probability to find
joint probability.
P( A  B)
P( A B) 
P( B)
P( A  B)  P( B) P( A B)
or
P( A  B)  P( A) P( B A)
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.6: The Multiplicative Rule
and Independent Events


Assume three of ten workers give illegal deductions
Event A: {First worker selected gives an illegal deduction}
Event B: {Second worker selected gives an illegal
deduction}


P(B|A) has only nine sample points, and two targeted
workers, since we selected one of the targeted workers in
the first round:


P(A) = P(B) = .1 + .1 + .1 = .3
P(B|A) = 1/9 + 1/9 = .11 + .11 = .22
The probability that both of the first two workers selected
will have given illegal deductions

P(AB) = P(B|A)P(A) = .(3) (.22) = .066
McClave: Statistics, 11th ed. Chapter 3:
Probability
34
3.6: The Multiplicative Rule
and Independent Events
McClave: Statistics, 11th ed. Chapter 3:
Probability
35
3.6: The Multiplicative Rule
and Independent Events
A Tree
Diagram
First selected
worker
Second
selected
worker
10 Workers
Illegal
Deductions
P = .3
Illegal
Deductions
P = .3 x.22
= .066
No Illegal
Deductions
P = .3 x.78
= .234
No Illegal
Deductions
P = .7
Illegal
Deductions
P = .7 x.33
= .231
McClave: Statistics, 11th ed. Chapter 3:
Probability
No Illegal
Deductions
P = .7 x .67
= .469
36
3.6: The Multiplicative Rule
and Independent Events

Dependent Events



P(A|B) ≠ P(A)
P(B|A) ≠ P(B)
Independent Events


 Studying stats 40
hours per week
 Working 40 hours
per week

Having blue eyes
P(A|B) = P(A)
P(B|A) = P(B)
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.6: The Multiplicative Rule
and Independent Events

Dependent
Events



P(A|B) ≠ P(A)
P(B|A) ≠ P(B)
Independent
Events


Mutually exclusive events are
dependent: P(B|A) = 0
P(A|B) = P(A)
P(B|A) = P(B)
Since P(B|A) = P(B),
P(AB) = P(A)P(B|A)
= P(A)P(B)
McClave: Statistics, 11th ed. Chapter 3:
Probability
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3.7: Random Sampling

If n elements are selected from a
population in such a way that every set
of n elements in the population has an
equal probability of being selected, the
n elements are said to be a (simple)
random sample.
McClave: Statistics, 11th ed. Chapter 3:
Probability
39
3.7: Random Sampling


How many five-card poker hands can be
dealt from a standard 52-card deck?
Use the combination rule:
 N   52   52! 
      
  2,598,960
 n   5   5!(52  5)!
McClave: Statistics, 11th ed. Chapter 3:
Probability
40
3.7: Random Sampling

Random samples can be generated by



Mixing up the elements and drawing by hand,
say, out of a hat (for small populations)
Random number generators
Random number tables


Table I, App. B
Random sample/number commands on
software
McClave: Statistics, 11th ed. Chapter 3:
Probability
41
3.8: Some Additional Counting
Rules
Situation
Number of Different Results
Multiplicative Rule

Draw one element from each
of k sets, sized n1, n2, n3, …
nk
Permutations Rule

Draw n elements, arranged
in a distinct order, from a set
of N elements
Partitions Rule

Partition N elements into k
groups, sized n1, n2, n3, … nk
(ni=N)
n1n2 n3    nk
N!
P 
( N  n)!
N
n
N!
n1! n2 ! n3!  nk !
McClave: Statistics, 11th ed. Chapter 3:
Probability
42
3.8: Some Additional Counting
Rules

Multiplicative Rule
Assume one professor from each of the six departments in a division
will be selected for a special committee. The various departments
have four, seven, six, eight, six and five professors eligible. How
many different committees, X*, could be formed?
X *  n1n2 n3    nk
 4 7  68 6 5
 87,360
McClave: Statistics, 11th ed. Chapter 3:
Probability
43
3.8: Some Additional Counting
Rules

Permutations
If there are eight horses entered in a race, how many different win,
place and show possibilities are there?
N!
P 
( N  n)!
8!

 336
(8  3)!
N
n
McClave: Statistics, 11th ed. Chapter 3:
Probability
44
3.8: Some Additional Counting
Rules

Partitions Rule
The technical director of a theatre has twenty stagehands. She
needs eight electricians, ten carpenters and two props people. How
many different allocations of stagehands, Y*, can there be?
N!
20!
Y* 

 8,314, 020
n1 !n2 !n3 ! nk ! 8!10!2!
McClave: Statistics, 11th ed. Chapter 3:
Probability
45
3.8: Bayes’s Rule

Given k mutually exclusive and exhaustive
events B1, B2,… Bk, and an observed event
A, then
P( Bi | A)  P( Bi
A) / P( A) 
P( Bi ) P( A | Bi )
.
P( B1) P( A | B1)  P( B 2) P( A | B 2)  P( Bk ) P( A | Bk )
McClave: Statistics, 11th ed. Chapter 3:
Probability
46
3.8: Bayes’s Rule


Suppose the events B1, B2, and B3, are mutually exclusive and
complementary events with P(B1) = .2, P(B2) = .15 and P(B3) = .65.
Another event A has these conditional probabilities: P(A|B1) = .4,
P(A|B2) = .25 and P(A|B3) = .6.
What is P(B1|A)?
P( B1 | A)  P( B1  A) / P( A)
P( B1) P( A | B1)
P( B1) P( A | B1)  P( B 2) P( A | B 2)  P( B3) P( A | B3)
.2  .4
.08
.08



 .158
.2  .4  .15  .25  .65  .6 .08  .0375  .39 .5075

McClave: Statistics, 11th ed. Chapter 3:
Probability
47