chapter 3: probability statistics. mcclave: statistics, 11th ed. chapter 3: probability 2 where...
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Chapter 3: Probability
Statistics
McClave: Statistics, 11th ed. Chapter 3: Probability
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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
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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
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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.
A four
A Head
An Ace
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
All probabilities must be between 0 and 1.
The probabilities of all the sample points must sum to 1.
Probability Rules for Sample Points
1
1i
n
i
p
0 1ip
McClave: Statistics, 11th ed. Chapter 3: Probability
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3.1: Events, Sample Spaces and Probability
All probabilities must be between 0 and 1
The probabilities of all the sample points must sum to 1
Probability Rules for Sample Points
1
1i
n
i
p
0 1ip
0 indicates an impossible outcome and 1 a certain outcome.
Something has to happen.
McClave: Statistics, 11th ed. Chapter 3: Probability
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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 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
If the number of sample points gets too large, we need a way to keep track of how they can be combined for different events.
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! 20 19 18 3 2 1
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 EventsMade of two or
more other events
UnionA B
Either A or B, or both, occur
IntersectionA B
Both A and Boccur
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 B
B C
A
B C
A C
A B 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
)(1)(
)(1)(
1)()(
APAP
APAP
APAP
C
C
C
McClave: Statistics, 11th ed. Chapter 3: Probability
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3.3: Complementary EventsA: {At least one head on two coin flips}
AC: {No heads}
43
41
41
43
41
41
41
1)(1)(
)(
)(
}{:
},,{:
C
C
C
APAP
AP
AP
TTA
THHTHHA
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:
)()()()( BAPBPAPBAP
McClave: Statistics, 11th ed. Chapter 3: Probability
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3.4: The Additive Rule and Mutually Exclusive Events
P(Surgery or Obstetrics) ( )
( ) ( ) ( ) ( )
( ) .12 .16 .02 .26
P A B
P A B P A P B P A B
P A B
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?
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
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3.4: The Additive Rule and Mutually Exclusive Events
)()()( BPAPBAP
If A and B are mutually exclusive,
3.5: Conditional Probability
Additional information or other events occurring may have an impact on the probability of an event.
26McClave: Statistics, 11th ed. Chapter 3: Probability
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.
27McClave: Statistics, 11th ed. Chapter 3: Probability
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).
28McClave: Statistics, 11th ed. Chapter 3: Probability
B BC
A A
3.5: Conditional Probability
29McClave: Statistics, 11th ed. Chapter 3: Probability
B BC
A A
)(
)()(
BP
BAPBAP
3.5: Conditional Probability
30McClave: Statistics, 11th ed. Chapter 3: Probability
B
A
)(
)()(
BP
BAPBAP
3.5: Conditional Probability
55% of sampled executives had cheated at golf (event A). P(A) = .55
20% of sampled executives had cheated at golf and lied in business (event B). 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)?
31McClave: Statistics, 11th ed. Chapter 3: Probability
3.5: Conditional Probability
P(A) = .55
P(A B) = .20 What is P(B|A)? )(
)()(
AP
ABPABP
364.55.
20.)( ABP
32McClave: Statistics, 11th ed. Chapter 3: Probability
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.
)(
)()(
BP
BAPBAP
)()()(
)()()(
ABPAPBAP
or
BAPBPBAP
33McClave: Statistics, 11th ed. Chapter 3: Probability
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(A) = P(B) = .1 + .1 + .1 = .3
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(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(AB) = P(B|A)P(A) = .(3) (.22) = .066
34McClave: Statistics, 11th ed. Chapter 3: Probability
McClave: Statistics, 11th ed. Chapter 3: Probability
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3.6: The Multiplicative Rule and Independent Events
3.6: The Multiplicative Rule and Independent Events
A Tree Diagram
First selected worker
Second selected worker
36McClave: Statistics, 11th ed. Chapter 3: Probability
3.6: The Multiplicative Rule and Independent Events
Dependent Events P(A|B) ≠ P(A) P(B|A) ≠ P(B)
Independent Events P(A|B) = P(A) P(B|A) = P(B)
Studying stats 40 hours per week
Working 40 hours per week
Having blue eyes
37McClave: Statistics, 11th ed. Chapter 3: Probability
3.6: The Multiplicative Rule and Independent Events
Dependent Events P(A|B) ≠ P(A) P(B|A) ≠ P(B)
Independent Events P(A|B) = P(A) P(B|A) = P(B)
Mutually exclusive events are dependent: P(B|A) = 0
Since P(B|A) = P(B),
P(AB) = P(A)P(B|A) = P(A)P(B)
38McClave: Statistics, 11th ed. Chapter 3: Probability
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.
39McClave: Statistics, 11th ed. Chapter 3: Probability
3.7: Random Sampling
How many five-card poker hands can be dealt from a standard 52-card deck?
Use the combination rule:
960,598,2)!552(!5
!52
5
52
n
N
40McClave: Statistics, 11th ed. Chapter 3: Probability
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
41McClave: Statistics, 11th ed. Chapter 3: Probability
3.8: Some Additional Counting Rules
Situation
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)
Number of Different Results
McClave: Statistics, 11th ed. Chapter 3: Probability
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!!!!
!
)!(
!
321
321
k
Nn
k
nnnn
N
nN
NP
nnnn
3.8: Some Additional Counting Rules
Multiplicative RuleAssume 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?
McClave: Statistics, 11th ed. Chapter 3: Probability
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360,87
568674
* 321
knnnnX
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?
McClave: Statistics, 11th ed. Chapter 3: Probability
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336)!38(
!8
)!(
!
nN
NPNn
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?
McClave: Statistics, 11th ed. Chapter 3: Probability
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1 2 3
! 20!* 8,314,020
! ! ! ! 8!10!2!k
NY
n n n n
3.8: Bayes’s Rule
Given k mutually exclusive and exhaustive events B1, B2,… Bk, and an observed event A, then
)
1 1 2 2
( | ) ( ) / ( )
( ) ( |.
( ) ( | ) ( ) ( | ) ( ) ( | )
i i
i i
k k
P B A P B A P A
P B P A B
P B P A B P B P A B P B P A B
46McClave: Statistics, 11th ed. Chapter 3: Probability
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)?
158.5075.
08.
39.0375.08.
08.
6.65.25.15.4.2.
4.2.
)|()()|()()|()(
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BAPBPBAPBPBAPBP
BAPBP
APABPABP
47McClave: Statistics, 11th ed. Chapter 3: Probability