Transcript File - VarsityField

3.1 Background(p88)
In everyday conversation, what does the
term “probability” measure?
I do not play Lotto … the
chances to win is too small
Mr president, the chances that
sales will decrease if we
increase prices are high
What is the chance that the
new investment will be
profitable?
How likely is it that the project
will be finished on time?
3.1 Background(p88)
“In everyday conversation, the term probability is a measure of one’s belief in
the occurrence of a future event”
NEW YORK, Mon: Mr. Webster Todd,
Chairman of the American National
Transportation Safety Board, said
today that the chances of two jumbo
jets colliding on the ground were about
6 million to one... –AAP
Professor Speed, who had strong
research interests in probability, was
intrigued by this statement and
wondered how the board had
calculated their figure. Speed wrote to
the chairman. In the reply it was stated
that the figure (6 million to one) has no
statistical validity nor was it intended to
be a rigorous probability statement.
3.1 Background(p88)
“…. Six million to one”
This is just a numerical measure of the very small likelihood of
the event to occur … called a probability.
Probability is a
numerical measure of the
likelihood that an event will occur
Impossible
0
Equal likely
Certain
1
3.1 Background (p88)
Deterministic versus Random experiment
3.1 Background(p88)
Statistics
Inferential
Statistics
Descriptive Statistics
Inference
Probability Theory
Randomness
3.1 Background(p88)
Experiment:
Outcomes:
Toss a coin
Select a part for inspection
Conduct a sales call
Roll a die
Play a football game
Head, Tail
Defective, Non-defective
Purchase, No-purchase
1, 2, 3, 4, 5, 6
Win, Lose, tie
Definition: event
An event is an outcome or a set of outcomes of a random experiment
Event (Capital letter) = { outcomes described by event}
e.g. the event “even number” when a die is rolled:
E = { 2, 4, 6 }
Probability is a numerical measurement of the likelihood that an
event will occur and is denoted as P(outcome)
3.1 Background(p88)
Experiment:
Outcomes:
Toss a coin
Select a part for inspection
Conduct a sales call
Roll a die
Play a football game
Head, Tail
Defective, Non-defective
Purchase, No-purchase
1, 2, 3, 4, 5, 6
Win, Lose, tie
Definition: sample space(p89)
The sample space is the set of ALL possible outcomes: S
e.g. S = {1, 2, 3, 4, 5, 6}
3.2 First Principles(p90)
Experiment: Roll a die
E1: Observe a 1
E2: Observe a 2
E3: Observe a 3
E4: Observe a 4
E5: Observe a 5
E6: Observe a 6
Simple events: They cannot be
decomposed - can have one and only
one sample point
Each outcome is equally likely
P(1) = P(2) = … P(6) = 1/6
P(outcome) = 1/N
N = total number of outcomes of the experiment
A= Observe an odd number
P(event) =
A  {E1, E3 , E5}  {1,3,5}
(# outcomes in the event)
N
P(A) = 3/6
3.2 First Principles(p90)
3.2 First Principles(p90)
3.2 First Principles(p90)
P(
)
Number of “pink” plants
Total number of plants
= 4/12 = 1/3
3.2 First Principles(p90): Example 3.1 (p91)
Subject
number
1
2
3
4
5
6
Gender
Age
M
F
M
F
M
F
40
42
51
58
67
70
Event
P(A) = 3/6
A = Female subjects
B = Male subjects
C = subjects over the age of 65
P(B) = 3/6
P(C) = 2/6
3.2 First Principles: Some rules and concepts(p91 – p95)
 Complement rule
The union of two events
The intersection of two events
The additional rule
Mutually exclusive
The conditional probability rule
Independent events
3.2 First Principles: Some rules and concepts(p91 – p95)
A graphic technique for visualizing
set
theory concepts using
overlapping circles and shading to
indicate intersection, union and
complement.
It was introduced in the late 1800s by
English logician, John Venn, although it is
believed that the method originated earlier.
3.2 First Principles: Some rules and concepts(p91 – p95)
Set:
“B”






Is an insect
Hatches from an egg
Compound eyes
Six legs
Two pairs of wings
Wings straight
above when at rest
 Thin hairless body
 Have a knob at the
end of the antennae
Elements of set B






Is an insect
Hatches from an egg
Compound eyes
Six legs
Two pairs of wings
Wings like a tent or
flat when at rest
 Wide furry body
 Antennae are thick
and furry
Elements of set M
Set:
“M”
•Is an insect
•Hatches from an egg
•Compound eyes
•Six legs
•Two pairs of wings
•Wings straight
above when at rest
•Thin hairless body
•Have a knob at the end
of the antennae
•Is an insect
•Hatches from an egg
•Compound eyes
•Six legs
•Two pairs of wings
•Wings like a tent or flat
when at rest
•Wide furry body
•Antennae are thick and
furry
3.2 First Principles: Some rules and concepts(p91 – p95)
•Wings straight
above when at
rest
•Thin hairless
body
•Have a knob
at the end of
the antennae
•Is an insect
•Hatches from
an egg
•Compound
eyes
•Six legs
•Two pairs of
wings
•Wings like a
tent or flat when
at rest
•Wide furry
body
•Antennae are
thick and furry
3.2 First Principles: Some rules(p91)
Subject
number
1
2
3
4
5
6
Gender
Age
M
F
M
F
M
F
40
42
51
58
67
70
Event
A = Female subjects
B = Male subjects
C = subjects over the age of 65
C
A
B
1
2
4
6
5
3
S
3.2 First Principles: Some rules(p91)
The Complement of an event:
= all outcomes in the sample space that are not in the event
P( A)  P( A )  1  P( A)
A
S
C
A
2
4
6
5
B
1
3
S
P (C )  4/6
3.2 First Principles: Some rules(p91)
The union of A and B is the event containing all sample
points belonging to A or B or both.
“At least one occurs”
A or B
The intersection of A and B is the event containing the
sample points belonging to both A and B.
“AND”
“Both events
occur”
3.2 First Principles: Some rules(p91)
C
A
2
4
6
5
B
1
3
S
P(Female or over the age of 65) = P( A  C ) 
4/6
P(Female and over the age of 65) = P( A  C )  1/6
P ( A  B)  ? A  B  
P ( )  0
Two events are said to be mutually
have no outcomes in common
exclusive if they
3.2 First Principles: Some rules(p91)
The additional rule
C
A
2
4
6
5
B
1
3
S
P( A  B)  P( A)  P( B)
P( A  C )  ?
P( A  C )  P( A)  P(C )  P( A  C )
3.2 First Principles: Some rules(p91)
Conditional probability
The probability of rain today (mid February) is 0.6
It has been raining the whole
week.
The probability of rain today
(mid February) ?????
3.2 First Principles: Some rules(p91)
Conditional probability
P(“1”) = 1/6
If we know that an odd number has fallen …
P(“1”) = 1/3
Conditional Probability
3.2 First Principles: Some rules(p91)
3.2 First Principles: Contingency Table or cross tabulation(p93)
80
70
50
70
30
150
List all possible outcome of the one event
List all possible outcome
of the other event
The sample space
3.2 First Principles: Contingency Table or cross tabulation(p93)
A two-way frequency distribution of 220 persons employed by a
specific research institution, classified according to type of post and
gender is given in the table below:
Calculate the probability that a randomly chosen employee:
a. Is male
P(M)=96/220
b. Is a female researcher
P(F  R ) 
80
220
c. is a female, given that the employee has a management post
3.2 First Principles: Contingency Table or cross tabulation(p93)
Educational level of patients seeking care at an allergy clinic
Gender
Male
Female
Total
Educational Level (years)
0-8
9 - 12 13 - 16 17 +
15
20
17
26
30
42
31
27
45
62
48
53
Total
78
130
208
Suppose a patient is selected at random, what is the probability that the patient
Is male?
78/208
Has 9 – 12 years of education?
62/208
Is female and has 9 – 12 years of education?
Has at most 12 years of education ?
42/208
P(0-8 or 9-12)=107/208
Is female if we know that the person only have between 9 – 12 years of
education?
42/62
3.2 First Principles: Some rules: Conditional probability (p94)
P( A  B)
P( A B) 
P( B)
Educational level of patients seeking care at an allergy clinic
Gender
Male
Female
Total
Educational Level (years)
0-8
9 - 12 13 - 16 17 +
15
20
17
26
30
42
31
27
45
62
48
53
Total
78
130
208
Is female if we know that the person only have between 9 – 12 years of
education? 42/62
P( Female  9  12)
P( Female 9  12) 
P(9  12)
42
42
208


62
62
208
3.2 First Principles: Independence (p95)
Two events are said to be independent if the occurrence of one
event does not influence the probability of the other
P( A B)  P( A)
P( B A)  P( B)
P( A  B)  P( A) P( B)
3.2 First Principles: Independent: Example 3.2B(p95)
Educational level of patients seeking care at an allergy clinic
Gender
Male
Female
Total
Educational Level (years)
0-8
9 - 12 13 - 16 17 +
25
30
25
25
25
30
25
25
50
60
50
50
Total
105
105
210
Are the two events “Male” and “17+” independent?
25
P( Male  17  ) 
210
 105  50  25
P( Male) P(17  )  


 210  210  210
Self study: Example 3.3
3.3 Combinations and permutations (p98)
1
2
3
Sampling With replacement
5
4
6
Sampling Without replacement
Order
important
Order not
important
Order
important
Order not
important
1
1
1
1
1
1
1
1
1
2
1
2
1
2
1
2
2
1
2
1
3.3 Combinations and permutations (p98)
Sampling With replacement
Order
important
Order not
important
Sampling Without replacement
Order
important
Order not
important
3.3 Combinations and permutations (p98)
ABC
ACB
BCA
BAC
CAB
CBA
1
2
3
3
1
2
ABC
3.3 Permutations (p99)
When sampling WITHOUT replacement, the number of distinct
arrangements (i.e., order important), called permutations of n
individuals from a population of N, is given by
N!
N Pn 
( N  n)!
N! = N(N-1)(N-2) … (3)(2)(1)0!
0! = 1
3.3 Combinations (p100)
When sampling WITHOUT replacement, the number of samples
in which order is not important, or combinations, of n
individuals from a population of size N is given by
N!
N Cn 
n!( N  n)!
3.3 Permutations and Combinations (p99)
3.3 Random Variable (p100)
Definition of a Random variable:
A random variable is a numerical description of the
outcome of an experiment
Experiment
Outcomes
Numerical Description = Random variable
3.3 Probability Distribution(p102)
1
3
2
4
Sampling Without replacement
1
0
1
1
2
1
0
1
1
2
0
1
1
2
5
6
Order not important
Let X = number of females
X
0
1
1
2
P(X)
3/15
9/15
3/15
1
The values of the random variable and the corresponding probabilities
constitutes a probability distribution
3.3 Probability Distribution(p102)
The probability distribution for a random variable describes how
probabilities are distributed over the values of the random
variable.
Consider the experiment of tossing a coin twice and noting
the outcome after every toss. Let X = the number of heads
The probability distribution of X:
1
2
X
H
0
H
T
1
2
T
H
T
T
H
P(X)
1/4
2/4
1/4
1
3.3 Probability Distribution(p102)
The probability distribution for a discrete variable Y can
be represented by a table or a graph or a formula.
The probability distribution of X:
H
T
H
T
H
H
T
T
X
P(X)
0
1
2
1/4
2/4
1/4
1
P(x)
0.6
2
P ( X  x)   (0.5) x (0.5) 2 x
 x
0.5
0.4
0.3
0.2
0.1
0
0
1
2
for x  0 ,1,2
3.3 Probability Distribution(p102)
A psychologist determined that the number of sessions
required to obtain the trust of a new patient is either 1, 2 or
3. Let X be a random variable indicating the number of
sessions required to gain the patient’s trust. The following
probability function has been proposed:
x
P( X  x) 
for x  1,2, or 3
6
a. Is this probability distribution valid? Explain.
b. What is the probability it takes exactly two sessions
to gain the patient’s trust?
c. What is the probability it takes at least two sessions
to gain the patient’s trust?