qt1 - 04 - probability

Introduction to Probability

Probability Theory

QUANTTECHINTEUQIASEVIT10S

Contents

Probability

Classical

Relative

Subjective

Marginal Probability

Joint Probability

Exclusive Events

Independent Events

Dependent Events

Bayes Theorem

Probability : Basic Terminology

Event

One or more of the possible outcomes of doing something

Toss a coin

Getting head

Getting tail

Person enters a mall

Person is male

Person is a student

Pick a carton of candies at a store

Candy is Amul

Candy is spoilt

Experiment

An activity that produces or causes an event is referred to as an experiment

Tossing a coin is an experiment.

Tossing three coins in a row is an experiment

Choosing a person at the entrance of a mall door and asking questions is an experiment

Pick a carton of candies at a store

Probability : Basic Terminology

Mutually Exclusive Events

Only one of the events can take place at a time

Either Head or Tail

Either Man or Woman

Non Exclusive Events

Both or more can take place

Person is male

Person is student

Collectively Exhaustive List

List of events that between them consider all cases

Toss of coin : Head or Tail nothing else is possible

Choice of cold drink

Coke

Pepsi

Anything else

Classical Probability

Probability of an event is defined as

[ number of outcomes where event occurs ]

[ total number of possible outcomes ]

Experiment of tossing a single coin

P(Head) = 1 / 2 = 0.5

Experiment of rolling a dice

P(getting 6) = 1 / 6 = 0.167

Experiment of choosing a person at the mall

P(Person is male ) = 0.5 [ unless we know more ]

P(Person is a student ) = not known as yet

Problem withClassicalProbability

State Space
All Possible Outcomes

HTHHHTTHTTHHHHHTHTTTHHHTHTHTTTHTTTEvent A : Head

Event B : Tail

Event A : Identical Reading

Event B : Different Reading

Event A : Three Heads

Event B : Two Heads

Event C : Anything Else

One CoinTossed

Two CoinsTossed

Three CoinsTossed

Probability is NOT certainty !

HTTTTTHTTH

P(H) = 0.5Probabilityof getting H = 0.5

HHTTTHHHTH

TTTTTHHTTT

THHHTHHTHT

HHHHHTHHHT

3/10

3/10

6/10

9/20

2/10

11/30

6/10

17/40

8/10

25/50

TTHHTTHTHT

4/10

29/60

HTHHTTHHHT

6/10

499/1000

Many

more

trIals

Simple Example

Classical Approach

I have three friends Ram, Gopal, Bharat each of whom is equally likely to visit me for lunch

What is the probability that today it will be Ram ?

Total 3 outcomes possible : R G B

One is the outcome in question : R

P(R) = 1/3

In the past one month

R has visited 20 times

G has visited 5 times

B has visited 5 times

What is the probability that today it will be Ram ?

Is it P(R) = 1/3

Relative Frequency

Probability calculated from statistical data on the relative frequency of certain occurrences

Observed relative frequency of an event in a very large number of trials

Proportion of times that an event occurs in the long run when conditions are stable

Carbon monoxide emissions

Challenges with relative probability

Number of events considered

Suppose you know of three of your friends who have had jaundice this year

Can you conclude that the probability of getting jaundice is higher this year than in the last year ?

Long run stability

Suppose Tata Motors brings a new model of Indica in the market and in the first three months 10% cars have a clutch failure.

Can you conclude that the probability of a clutch failure in Indica cars is 0.1 ?

If you toss a coin

Two times you may get two heads

100 times, you would get between 45 55 heads

Two Questions

A yoga club consists of 10 members of whom

2 Doctors, 3 Engineers, 3 CA and 2 Other

4 women, 6 men

7 married, 3 single

If we were to choose a club secretary by lottery, what is the probability that the secretary is a

Doctor ?

Woman ?

Single person ?

The following table shows the frequency of marks in a quiz

If I were to ask a random student what is your score, what is the probability that it is in the range of

2 3 ?

6 7 ?

Points to note

Numerical value of a probability of any event is between 0 and 1

0 : if it is impossible, e.g. The probability of teacher being an alien from outer space

1 : if is totally certain, e.g. The probability of teacher being a human being

Sum of the probabilities of events that must be 1, provided the events are

Mutually exclusive

Part of a collectively exhaustive list

Something must happen !!

Subjective Probability

Based on beliefs, not hard facts ..

Because hard facts are simply not available and you cannot sit and do nothing because you do not have facts

Widely used in cases where the event in question occurs very rarely

To be replaced with statistical models if possible

Consider

Selection of a candidate for a job

Sanction of a loan

Earlier these were all subjective but now with more data being collected

Strategic Decisions are still taken on the basis of subjective probability !!


Relative Probability


[unconditional probability]

Probability of an event is written as

P(A) = n where

A is the event that secretary is

Doctor

Engineer

CA

Other

n is the value of the probability 0 Joint Probability
under Statistical Dependence

P(Man/Employed)

P(Man AND Employed)

P(Employed)

P(A / B)

P(A B)

P(B)

P(Man and Employed)

= P(Man / Emp) x P(Emp)

P (A B ) = P (A / B ) x P(B)

=

=

Joint Probabilities
from Conditional Probabilities

P(E M) = P(E/M) x P(M) = 3/5 x 5/10 = 3/10

= P(M/E) x P(E) = 3 / 4 x 4/10 = 3/10

P(E W) = P(E/W) x P(W) = 1/5 x 5/10 = 1/10

P(S M) = P(S/M) x P(M) = 2/5 x 5/10 = 2/10

P(S W) = P(S/W) x P(W) = 4/5 x 5/10 = 4/10

Probabilities under
Statistical Dependence

Conditional ProbabilityP(B/A) = P(B A) / P(A)P(A/B) = P(B A) / P(B)

Joint ProbabilityP(B A) = P(B / A) x P(A)= P(A / B) X P(B)

Where are we ? 3




Marginal Probability P(A)

Probability of A or B

Exclusive Events P(AorB) = P(A) + P(B)

Non Exclusive Events P(AorB) = P(A)+P(B)-P(AB)

Two Events under Statistical Independence

Joint probability P(AB) = P(A) x P(B)

Conditional Probability P(B/A) = P(B)

Two Events under Statistical Dependence

Conditional Probability P(B/A) = P(AB) / P(A)

Joint Probability P(AB) = P(B/A) x P(A)

Marginal Probabilities

Marginal Properties under statistical dependence are computed by simply summing up the probabilities of all the joint events where the simple event occurs

P(A) = P(AB) + P(AC)

P(Man) = P(Man Employed) + P(Man Student)

P(Student) = P(Man Student) + P(Woman Student)

Marginal Probabilities

P(M) = P( E M) + P(S M) = 0.3 + 0.2 = 0.5

P(W) = P(E W) + P(S W) = 0.1 + 0.4 = 0.5

P(E) = P(E M) + P(E W) = 0.3 + 0.1 = 0.4

P(S) = P(S M) + P(S W) = 0.2 + 0.4 = 0.6

Where are we ? 4




Marginal Probability P(A)

Probability of A or B

Exclusive Events P(AorB) = P(A) + P(B)

Non Exclusive Events P(AorB) = P(A)+P(B)-P(AB)

Two Events under Statistical Independence

Joint probability P(AB) = P(A) x P(B)

Conditional Probability P(B/A) = P(B)

Two Events under Statistical Dependence

Conditional Probability P(B/A) = P(AB) / P(A)

Joint Probability P(AB) = P(B/A) x P(A)

Multiple Events under Statistical Dependence

Marginal Probality P(A) = P(AB) + P(AC) + P(AD) + .....

Bayes Theorem
revising prior estimates of probability

Central Theorem that connects

Joint Probability

Conditional Probability


Using the basic formula

P(B A) = P(B/A) x P(A)

P(B A) = P(A/B) x P(B)

P(B/A) x P(A) = P(A/B) x P(B)

Its value lies in being able to calculate P(B/A) from the value of P(A/B)

Brand Conscious Customer

Suppose there are two types of customers

Type A : Brand Conscious; Type B : Brand Indifferent

Demographic information tells us that 20% customers are Type A and rest 80% is Type B

P(A) = 0.2 ; P(B) = 0.8

P(A) + P(B) = 1 since events are mutually exclusive

Probability of sale of Designer Shirt is

60% if the customer is Brand Conscious

10% if the customer is Brand Indifferent

P(S/A) = 0.6, P(S/B) = 0.1

A person walks in and purchases a Designer Shirt

What is the probability that he is Brand Conscious

Calculate P(A/S)

Brand Conscious Customer : 1

Demographics tells us that P(A) = 0.2; P(B) = 0.8We know the conditional probabilitiesP(S/A) = 0.6, P(S/B) = 0.1Joint Probability of Sale and given type of customer hence P(SA) = P(S/A)xP(A)Marginal Probability of Saleis sum of the two joint probabilitiesP(S) = P(SA) + P(SB)


Conditional ProbabilityP(A/S) = P(SA) / P(S)P(B/S) = P(SB) / P(S)

After purchase of ONE branded shirt

Probability that customer is brand conscious has now increased from 20% to 60%

a second sale

After purchase of TWO branded shirts

Probability that customer is brand conscious has now increased from 20% to 90%

Consider Loyalty Program / Free Gifts !!

The TWO sales are INDEPENDENT of each otherhence P(2 Sales) = P(1 Sale) x P(1 Sale)

Error in Machine Setup

A carton sealing machine has to be aligned correctly before it starts packaging

10% of the time it is aligned badly : P(B) = 0.1

90% of the time it is aligned correctly : P(C) = 0.9

If the machine is aligned is aligned badly then there is a 50% chance that the carton will be found defective by the QC staff

P(D / B ) = 0.5

If the machine is aligned correctly, the probability of a defect is is 5%

P(D/C) = 0.05

Given that the first carton is rejected, what is the probability that alignment is bad ? What is P(B/D)

Error in Machine Setup : 1

After the detection of ONE defect on the first carton

Probability that alignment is erroneous has now increased from 10% to 53%

Let us rework the numbers such that

Probability of BAD alignment = 2%

Probability of Defective carton with

Bad alignment is 5%

Good alignment is 1%

Is one defective carton important enough ?


After the detection of ONE defect on the first carton

Probability that alignment is erroneous has now increased from 2% to 9%

Is it worth stopping production and fixing alignment ?

Let us consider 2 defects in the first 5 cartons

inconsistent behaviour

After TWO defects in FIVE cartons

Probability that machine is badly aligned has now increased from 2% to 31%

Consider Stopping Production and fixing the machine

P(2 D in 5) = 0.05 x 0.05 X 0.95 X 0.95 X 0.95 = 0.00214P(2D in 5) = 0.01 x 0.01 x 0.99 x 0.99 x 0.99 = 0.00010

Customer Type Elementary Event(Prior) Probability of Primary EventConditional Probability of ONE Sale, given Customer TypeConditional Probability of TWO Sales, given Customer TypeJoint Probability of Sale and Customer Type(Posterior) Probability of Primary Event given 1 Sale

Brand Conscious Type A0.20.60.360.2 x 0.36 =0.0720.9

Brand Indifferent Type B0.80.10.010.8 x 0.1 =0.0080.1

Marginal Probability of Sale =>0.080

???Page ??? (???)30/05/2008, 10:20:30Page / Alignment Elementary Event(Prior) Probability of Primary EventConditional Probability of Defect, given Alignement TypeJoint Probability of Defect and Alignment Type(Posterior) Probability of Primary Event given 1 Defect

BAD Type B0.10.50.1 x 0.5 =0.0500.53

Correct Type C0.90.050.9 x 0.05 =0.0450.47

Marginal Probability of Defect =>0.095

???Page ??? (???)30/05/2008, 10:20:30Page / Alignment Elementary Event(Prior) Probability of Primary EventConditional Probability of Defect, given Alignement TypeJoint Probability of Defect and Alignment Type(Posterior) Probability of Primary Event given 1 Defect

BAD Type B0.020.050.1 x 0.5 =0.0010.09

Correct Type C0.980.010.9 x 0.05 =0.0100.91

Marginal Probability of Defect =>0.011

???Page ??? (???)30/05/2008, 10:20:30Page / Alignment Elementary Event(Prior) Probability of Primary EventConditional Probability of ONE Defect, given Alignement TypeConditional Probability of 2 Defects in 5 CartonsJoint Probability of Sale and Customer Type(Posterior) Probability of Primary Event given 1 Sale

BAD Type B0.020.050.002140.02 x 0.002140.0000430.31

Correct Type C0.980.010.000100.98x0.00010.0000950.69

Marginal Probability of Defect =>0.0001380.050.002140.950.010.000100.99

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???Page ??? (???)27/05/2008, 22:11:08Page / DECOWWWWMMMMMM

???Page ??? (???)27/05/2008, 22:11:08Page / Event / PersonProbability of Event P(n)Description of Event / Kind of Person

10.1employed man

20.1employed man

30.1employed man

40.1employed woman

50.1student man

60.1student man

70.1student woman

80.1student woman

90.1student woman

100.1student woman

???Page ??? (???)29/05/2008, 15:54:06Page / EventProbabilityDescription of Event Work

10.1employed manemployed



40.1employed womanemployed

50.1student manstudent

60.1student manstudent

70.1student womanstudent




???Page ??? (???)29/05/2008, 20:06:15Page / EventProbabilityDescription of Event WorkGender

10.1employed manemployedman



40.1employed womanemployedwoman

50.1student manstudentman


70.1student womanstudentwoman


























???Page ??? (???)29/05/2008, 18:09:01Page / Customer Type Elementary Event(Prior) Probability of Primary EventConditional Probability of Sale, given Customer TypeJoint Probability of Sale and Customer Type

Brand Conscious Type A0.20.60.2 x 0.6 =0.12

Brand Indifferent Type B0.80.10.8 x 0.1 =0.08


???Page ??? (???)30/05/2008, 10:20:30Page / Customer Type Elementary Event(Prior) Probability of Primary EventConditional Probability of Sale, given Customer TypeJoint Probability of Sale and Customer Type(Posterior) Probability of Primary Event given 1 Sale

Brand Conscious Type A0.20.60.2 x 0.6 =0.120.6

Brand Indifferent Type B0.80.10.8 x 0.1 =0.080.4


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qt1 - 04 - probability

Education