Transcript Chapter 3
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 4
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 5
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 6
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 7
3.1: Events, Sample Spaces and Probability Probability Rules for Sample Points All probabilities must be between 0 and 1.
0
p i
1 The probabilities of
all
the sample points must sum to 1.
i n
1
p i
1 McClave: Statistics, 11th ed. Chapter 3: Probability 8
3.1: Events, Sample Spaces and Probability Probability Rules for Sample Points All probabilities must be between 0 and 1 0
p i
1
0 indicates an impossible outcome and 1 a certain outcome.
The probabilities of
all
the sample points must sum to 1
Something has to happen.
i n
1
p i
1 McClave: Statistics, 11th ed. Chapter 3: Probability 9
3.1: Events, Sample Spaces and Probability An
event
points: is a specific collection of sample Event
A
: Observe an even number.
McClave: Statistics, 11th ed. Chapter 3: Probability 10
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 11
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 12
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 The sum of the sample point probabilities is the event probability McClave: Statistics, 11th ed. Chapter 3: Probability 13
3.1: Events, Sample Spaces and Probability If a sample of
n
of
N
elements is drawn from a set elements (
N ≥ n),
the number of different samples is
N n
N
!
n
)!
.
McClave: Statistics, 11th ed. Chapter 3: Probability 14
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 5 20!
3 2 1 3 2 1 15,504.
McClave: Statistics, 11th ed. Chapter 3: Probability 15
3.2: Unions and Intersections Compound Events
Made of two or more other events
Union
A
B Either A or B, or both, occur
McClave: Statistics, 11th ed. Chapter 3: Probability Intersection
A
B Both A and B occur
16
3.2: Unions and Intersections McClave: Statistics, 11th ed. Chapter 3: Probability 17
3.2: Unions and Intersections A A B B A B C B C A C C McClave: Statistics, 11th ed. Chapter 3: Probability 18
3.3: Complementary Events The
complement
event that
A
of any event
A
does not occur,
A C
.
is the A: {Toss an even number} A C : {Toss an odd number} B: {Toss a number ≤ 3} B C : {Toss a number ≥ 4} A [A B = {1,2,3,4,6} B] C = {5}
(Neither A nor B occur)
McClave: Statistics, 11th ed. Chapter 3: Probability 19
3.3: Complementary Events
P
(
A
)
P
(
A C
) 1
P
(
A
) 1
P
(
A C
)
P
(
A C
) 1
P
(
A
) McClave: Statistics, 11th ed. Chapter 3: Probability 20
3.3: Complementary Events A: {At least one head on two coin flips} A C : {No heads}
A
: {
HH
,
HT
,
TH
}
A C
: {
TT
}
P
(
A
) 1 4 1 4 1 4 3 4
P
(
A C
) 1 4
P
(
A
) 1
P
(
A C
) 1 1 4 3 4 McClave: Statistics, 11th ed. Chapter 3: Probability 21
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 22
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)
B B
) ) .26
B
)
B
) McClave: Statistics, 11th ed. Chapter 3: Probability 23
3.4: The Additive Rule and Mutually Exclusive Events Events
A
exclusive
and
B
if
A
are
mutually
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 27
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 B C within B C ).
(including the A region B B C A A McClave: Statistics, 11th ed. Chapter 3: Probability 28
3.5: Conditional Probability
P
(
A B
)
P
(
A
B
)
P
(
B
) B A A B C McClave: Statistics, 11th ed. Chapter 3: Probability 29
3.5: Conditional Probability
P
(
A B
)
P
(
A
B
)
P
(
B
) B A McClave: Statistics, 11th ed. Chapter 3: Probability 30
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)?
McClave: Statistics, 11th ed. Chapter 3: Probability 31
3.5: Conditional Probability P(A) = .55
P(A B) = .20
What is P(B|A)?
P
(
B A
)
P
(
B
A
)
P
(
A
)
P
(
B A
) .
20 .
55 .
364 McClave: Statistics, 11th ed. Chapter 3: Probability 32
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 33
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(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
No Illegal Deductions P = .7
Illegal Deductions P = .3 x.22
= .066
No Illegal Deductions P = .3 x.78
= .234
Illegal Deductions P = .7 x.33
= .231
No Illegal Deductions P = .7 x .67
= .469
McClave: Statistics, 11th ed. Chapter 3: Probability 36
3.6: The Multiplicative Rule and Independent Events Dependent Events P(A|B) ≠ P(A) P(B|A) ≠ P(B) Studying stats 40 hours per week Working 40 hours per week Independent Events P(A|B) = P(A) P(B|A) = P(B) Having blue eyes McClave: Statistics, 11th ed. Chapter 3: Probability 37
3.6: The Multiplicative Rule and Independent Events Dependent Events P(A|B) ≠ P(A) P(B|A) ≠ P(B) Mutually exclusive events are dependent: P(B|A) = 0 Independent Events 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 38
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 n
52 5 52 5 !
( 52 !
5 )!
2 , 598 , 960 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
Multiplicative Rule Draw one element from each of
k
sets, sized
n 1 , n 2 , n 3 , … n k
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
n 1 , n 2 , n 3 , … n k
(
n i =N
)
Number of Different Results
n P
1
n n N
2
n
3
n k
(
N N
!
n
)!
N
!
n
1 !
n
2 !
n
3 !
n k
!
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
*
n
1
n
2
n
3
n k
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?
P n N
(
N N
!
n
)!
8 !
( 8 3 )!
336 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?
Y
* !
2 !
N
!
3 !
n k
!
20!
8!10!2!
8, 314, 020 McClave: Statistics, 11th ed. Chapter 3: Probability 45
3.8: Bayes’s Rule Given
k
mutually exclusive and exhaustive events
B 1
,
A
, then
B 2
, …
B k
, and an observed event
i
| )
i A P A
1 ) ( | 1 )
P B P A B i
) 2 ) ( | 2 )
k
) ( |
k
) .
McClave: Statistics, 11th ed. Chapter 3: Probability 46
3.8: Bayes’s Rule Suppose the events
B 1 , B 2 ,
and P(
A
|
B 2
) = .25 and P(
A
|
B 3
) = .6.
B 3 ,
are mutually exclusive and complementary events with P(
B 1
) = .2, P(
B 2
) = .15 and P(
B 3
) = .65. Another event
A
has these conditional probabilities: P(
A
|
B 1
) = .4, What is P(
B 1
|
A
)?
P
(
B
1 |
A
)
P
(
B
1
A
) /
P
(
A
)
P
(
B
1 )
P
(
A
|
B
1 )
P
(
B
1 )
P
(
A
|
B
1 )
P
(
B
2 )
P
(
A
|
B
2 )
P
(
B
3 )
P
(
A
|
B
3 ) .
2 .
4 .
.
2 15 .
4 .
25 .
65 .
6 .
08 .
08 .
0375 .
39 .
08 .
5075 .
158 McClave: Statistics, 11th ed. Chapter 3: Probability 47