i do not play lotto … the chances to win is too small in everyday conversation, what does the term...

I do not play Lotto … the chances to win is too small

In everyday conversation, what does the term “probability” measure?

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.

0

Impossible

1

CertainEqual likely

Probability is a

numerical measure of the

likelihood that an event will occur

3.1 Background(p88)

Deterministic versus Random experiment

3.1 Background (p88)

Statistics

Descriptive Statistics Inferential Statistics

3.1 Background(p88)

InferenceProbability Theory

Randomness

Experiment:

Toss a coinSelect a part for inspectionConduct a sales callRoll a diePlay a football game

Outcomes:

Head, TailDefective, Non-defectivePurchase, No-purchase1, 2, 3, 4, 5, 6Win, Lose, tie

An event is an outcome or a set of outcomes of a random experiment

Definition: event

e.g. the event “even number” when a die is rolled: E = { 2, 4, 6 }

Event (Capital letter) = { outcomes described by event}

3.1 Background(p88)

Probability is a numerical measurement of the likelihood that an event will occur and is denoted as P(outcome)

Experiment:

Toss a coinSelect a part for inspectionConduct a sales callRoll a diePlay a football game

Outcomes:

Head, TailDefective, Non-defectivePurchase, No-purchase1, 2, 3, 4, 5, 6Win, Lose, tie

The sample space is the set of ALL possible outcomes: S

Definition: sample space(p89)

e.g. S = {1, 2, 3, 4, 5, 6}

3.1 Background(p88)

E1: Observe a 1

E2: Observe a 2

E3: Observe a 3

A= Observe an odd number

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

}5,3,1{},,{ 531 EEEA

Experiment: Roll a die

3.2 First Principles(p90)

P(outcome) = 1/N

N = total number of outcomes of the experiment

P(1) = P(2) = … P(6) = 1/6

P(event) = (# outcomes in the event)

NP(A) = 3/6

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)

3.2 First Principles(p90): Example 3.1 (p91)

Subject number

Gender Age

1 M 40 2 F 42 3 M 51 4 F 58 5 M 67 6 F 70

Event

A = Female subjectsB = Male subjectsC = subjects over the age of 65

P(A) = P(B) = P(C) =

3/6

3/6

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

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)

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

Set:“B”

Set:“M”

Elements of set B Elements of set M

3.2 First Principles: Some rules and concepts(p91 – p95)

•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

•Wings straight above when at rest•Thin hairless body•Have a knob at the end of the antennae 

•Wings like a tent or flat when at rest  •Wide furry body•Antennae are thick and furry

•Is an insect•Hatches from an egg•Compound eyes•Six legs•Two pairs of wings 

3.2 First Principles: Some rules and concepts(p91 – p95)

3.2 First Principles: Some rules(p91)

Subject number

Gender Age

1 M 40 2 F 42 3 M 51 4 F 58 5 M 67 6 F 70

Event

A = Female subjectsB = Male subjectsC = subjects over the age of 65

2

46

A1

35

BC

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

S

)(1)()( APAPAP A

24

6

A 1

35

BC

S

)(CP 4/6

“At least one occurs”

A or B

The union of A and B is the event containing all sample points belonging to A or B or both.

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)

3.2 First Principles: Some rules(p91)

24

6

A 1

35

BC

S

4/6P(Female or over the age of 65) = )( CAP

P(Female and over the age of 65) = )( CAP 1/6

?)( BAP0)(

P

BA

Two events are said to be mutually exclusive if they have no outcomes in common

3.2 First Principles: Some rules(p91)

The additional rule

24

6

A 1

35

BC

S

)()()( BPAPBAP

)()()()( CAPCPAPCAP

?)( CAP

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

1/6

If we know that an odd number has fallen …

P(“1”) = 1/3

Conditional Probability

P(“1”) =

3.2 First Principles: Some rules(p91)

Conditional probability

3.2 First Principles: Some rules(p91)

150

80

70

50 70 30

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

b. Is a female researcher

c. is a female, given that the employee has a management post

P(M)=96/220

P F R( ) 80

220

3.2 First Principles: Contingency Table or cross tabulation(p93)

Educational Level (years) Gender 0 - 8 9 - 12 13 - 16 17 + Total Male 15 20 17 26 78 Female 30 42 31 27 130 Total 45 62 48 53 208

Educational level of patients seeking care at an allergy clinic

Suppose a patient is selected at random, what is the probability that the patient

Is male? Has 9 – 12 years of education?

Is female and has 9 – 12 years of education?

Has at most 12 years of education ?

Is female if we know that the person only have between 9 – 12 years of education?

62/20878/208

P(0-8 or 9-12)=107/208

42/62

42/208

3.2 First Principles: Contingency Table or cross tabulation(p93)

)(

)()(

BP

BAPBAP

Educational Level (years) Gender 0 - 8 9 - 12 13 - 16 17 + Total Male 15 20 17 26 78 Female 30 42 31 27 130 Total 45 62 48 53 208

Educational level of patients seeking care at an allergy clinic

Is female if we know that the person only have between 9 – 12 years of education? 42/62

)129(

)129(

P

FemaleP)129( FemaleP

62

42

20862

20842

3.2 First Principles: Some rules: Conditional probability (p94)

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

)()()(

)()(

)()(

BPAPBAP

BPABP

APBAP

Educational Level (years) Gender 0 - 8 9 - 12 13 - 16 17 + Total Male 25 30 25 25 105 Female 25 30 25 25 105 Total 50 60 50 50 210

Educational level of patients seeking care at an allergy clinic

Are the two events “Male” and “17+” independent?

3.2 First Principles: Independent: Example 3.2B(p95)

210

25

210

50

210

105)17()(

210

25)17(

PMaleP

MaleP

Self study: Example 3.3

1 3 52 4 6

3.3 Combinations and permutations (p98)

Sampling With replacement Sampling Without replacement

Order important

Order not important

Order important

Order not important

1 1

1 2

2 1

1 1

1 2

1 1

1 2

2 1

1 1

1 2

3.3 Combinations and permutations (p98)

Sampling With replacement

Sampling Without replacement

Order important

Order not important

Order important

Order not important

1

23

1 23

3.3 Combinations and permutations (p98)

ABCACBBCABACCABCBA

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

)!(

!

nN

NPnN

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

)!(!

!

nNn

NCnN

3.3 Permutations and Combinations (p99)

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 Random Variable (p100)

3.3 Probability Distribution(p102)

Sampling Without replacement Order not important

1 3 52 4 6

Let X = number of females

X P(X)

0

1

2

3/15

1

9/15

3/15

1 1 1

1

0 0

1

1 1

1

1

0

2

22

The values of the random variable and the corresponding probabilities constitutes a probability distribution

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:

X P(X)

0

1

2

1/4

1

2/4

1/4

H H

H T

T

T T

H

3.3 Probability Distribution(p102)

1 2

The probability distribution of X:

X P(X)

012

1/4

1

2/41/4

H HH TTT T

H

0 1 20

0.1

0.2

0.3

0.4

0.5

0.6

P(x)

210for )5.0()5.0(2

)( 2 ,,xx

xXP xx

The probability distribution for a discrete variable Y can be represented by a table or a graph or a formula.

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:

3 ,2,1for 6

)( orxx

xXP

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?

3.3 Probability Distribution(p102)