DNSC6221_Lecture2

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DNSC 6221

JUD

Lecture Set 02Bumsoo Kim

05/29/2012

The George Washington University School of Business

Department of Decision Sciences

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Introduction to Probability Brief history of probability

Probability interpretations

Classical approach

Relative frequency approach

Subjective approach

Definition of probability

Laws of probability (basic principles)

Conditional probability

Statistical independence and mutually exclusive events

Bayes Theorem

Probability trees

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Brief History of Probability

The use of probability as a concept to measure uncertaintydates back to hundreds of years ago. Probability has

found applications in areas as diverse as medicine,gambling, weather forecasting and the law.

It is generally believed that the mathematical theory ofprobability was started by the French mathematician

Blaise Pascal (1623-1662) and Pierre Fermat (1601-1665)when they succeeded in deriving exact probabilities forcertain gambling problems involving dice.

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Probability Interpretations1. Frequency Approach:

The probability of a particular outcome can be obtained if the

process is repeated a large number of times under similar

conditions.Ex: tossing a coin

2. Classical Approach:

The classical approach is a mathematical formula based

approach in which the possible outcomes are enumerated

and exact probabilities are calculated.

Ex: rolling two dice

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Probability Interpretations The classical approach would not work in processes where

we cannot describe the possible outcomes and there is no

way to determine the probabilities mathematically.Ex: Whether or not a project will be successful

3. Subjective Approach:

According to the subjective, or personal, interpretation ofprobability, the probability that a person assigns to a possible

outcome of some process represents her own judgment of the

likelihood that the outcome will be obtained (degree of belief).

Ex: Expert opinions

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Basic Principles Note that the calculus of probability applies equally well no

matter which interpretation one prefers.

Likelihood, chance

Only considerrandom experiments

Characteristics of random experiments

Can specify all possible outcomes.

Cannot predict a specific outcome with certainty.

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Basic Principles Ex:Todays closing price change of a security relative to

yesterday.

Probability is the yardstick for measuring the uncertainty of these

outcomes.

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Venn Diagram of Outcomes

- Tool to represent probabilities

- Used in set theory to show the mathematical orlogical relationship between different sets

Outcomes of a roll of a die

Sample space: {1, 2, 3, 4, 5 ,6}Event A: outcome is even: {2, 4, 6}

Event B: outcome is > 3: {4, 5, 6}


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Basic PrinciplesExample:

An automobile dealer sells two brands of new cars. One, C, is primarilyAmerican in origin; the other, G, is primarily Japanese. The dealerperforms repair work under warranty for both brands. Each warrantyjob is classified according to the primary problem to be fixed. If there ismore than one problem in a given job, all problems are listedseparately. Records for the past year indicate the following numbers ofproblems:

Problem AreaEngine Transmission Exhaust Fit/Finish Other Total

Brand C 106 211 67 133 24 541

G 21 115 16 24 6 182Total 127 326 83 157 30 723

a. What is the probability that a randomlychosen problem comes frombrand C?P( randomly chosen problem comes from brand C) = .748

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Basic Principles

A B

Venn diagram

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Mutually exclusive events: Only one of many events can

occur at the same time


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Basic PrinciplesExample:


Brand C 106 211 67 133 24 541G 21 115 16 24 6 182Total 127 326 83 157 30 723

b. Serious problems are those involving the engine or transmission.What is the probability that a randomly chosen problem is serious?

P (Serious problem) = P(E) + P(T)= 127/723 + 326/723= 453/723= .627

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Addition Law of Probability General Addition Law

P(A or B) = P(A) + P(B) P(A and B)

AB (A and B) has been counted twice

Subtract once

Example:

What is the probability that a randomly chosen problem comes from Brand

C or is an engine problem?

C denotes "Brand C. P(C) = 541/723

E denotes Engine problem". P(E) =127/723

P(C or E) = P(C) + P(E) P(C and E)

= 541/723 + 127/723 - 106/723= 562/723 = .777 12


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Complements Law

A A

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A` is also used to represent the complement of A.


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Complements Law Ex:Birthday Problem

- There are n people in a room. What is the probability that at least twoof them have the same birthday?

Assumptions: No birthdays fall on Feb 29;

All 365 days are equally likely for each person; and

Birthdays are independent

A = {at least two people have same birthday}

A` ={no two people have the same birthday}P (A` ) = (365)(364)(363)[365 (n-1)] / (365)n

= .29 for n=30 so P(A)= 0.71for n=50 P(A)=.97

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Conditional vs Marginal

Probability Conditional Probability

Conditional probability of event A occurring given that event B hasoccurred, denoted by P(A|B), is:

P(A|B) = P(A and B) / P(B),provided P(B) > 0.

A B

Aand B

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Marginal Probability: Probability of one variable taking a valueirrespective of others, i.e; P(A), P(B)


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Conditional vs Marginal

ProbabilityExample:


Brand C 106 211 67 133 24 541

G 21 115 16 24 6 182Total 127 326 83 157 30 723

a. For the auto dealers data, what is the probability that a randomly chosen problem is an engineproblem, given that it comes from brand C?

P(Engine problem | Brand C) = P(E|C) = P(E and C)/P(C)= (106/723)/(541/723)

=.196where P(Brand C and Engine Problem) = P(C and E) = 106/723and P(Brand C) = P (C) = 541/723

Note: P (E|C) = .196 while P(E) = .176The occurrence of event C affects P(E)

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Multiplication Rule Multiplication Rule

From the definition of conditional probability,

P(A and B) = P(B) P(A|B)

Obviously,P(A and B) = P(A) P(B|A)

Example: Randomly pick (without replacement) 2 cardsfrom a standard deck. Find probability of 2 hearts.

A = {1st

card is a heart}, B = {2nd

card is a heart}P (A and B) = P(A) P(B|A) = (13/52) (12/51)

The multiplication rule is useful in Probability Trees.

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Statistical Independence If occurrence of one event does not affect the other, they are

independent events.

Mathematically,A and B are independent events if and only if

P(A|B) = P(A)

Example:

Are events Brand C and Engine Problem independent?

P(Engine Problem) = P(E) = 127/723 = .176

P(Engine Problem|Brand C) = P(E|C) = .196

Conclusion:The events Brand C and Engine Problemare notindependent because

P(E|C) P(E)

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Statistical Independence Multiplication Rule for Independent Events

If A and B are independent, then

P(A and B) = P(A) P(B)

Reason: From Multiplication Rule,

P(A and B) = P(A|B) P(B)

From independence,P(A|B) = P(A)

P(A and B) = P(A) P(B)

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Independent vs Mutually

Exclusive Events Coin flip example:

One flip: Heads or Tails are mutually exclusive eventsTwo flips: The outcome of each flip is independent

If A and B are mutually exclusive events, then A and B can't bothhappen at the same time. Mutually exclusive events have nobasic outcomes in common: P( A and B) = 0

If A and B are independent, then P(A and B) = P(A) P(B)

Mutually exclusive events are always dependent.

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Some Useful Definitions

Union of EventsThe union of two events A and B is all basic

outcomes that are part of either A or B or both:A B = A or B

Intersection of Events (Joint Probability)

The intersection of two events A and B is all basicoutcomes that are part of both A and B:

A B = A and B

P( A and B) = P( A , B)


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Formulas so far Definition: 0


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ExampleExample :

Suppose that A and B are mutually exclusive events.

P(A) =0.4 and P(B) =0.3

Find P(A and B), P(A or B), P(not A and not B), P(A and not

B), P(A|B), P(A|not B) and P(B|not A)

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Probability Tables

- Tool to compute and represent probabilities

- Ex: Bivariate Joint Probabilities

B1 B2 . . . Bk

A1 P(A1 B1) P(A1 B2) . . . P(A1 Bk)

A2

P(A2 B1) P(A2 B2) . . . P(A2 Bk)

. . . . .

Ah

P(Ah B1) P(Ah B2) . . . P(Ah Bk)


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Probability Trees

Many problems require the successive use of several of the basicprinciples to obtain a solution. These problems can be solvedalgebraically, but it is helpful to be familiar with some tools thatcan help you keep the logic straight. One of these tools isprobability (or decision) trees.

Probability trees use the multiplication law of probabilities.

In a probability tree, the probability for a specific path is found byusing the multiplication rule.

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Probability TreesExercise:

A purchasing dept. finds that 75% of its special orders are

received on time. Of those orders that are on time, 80% meetspecifications completely; of those orders that are late, 60%

meet them.

T = {Order is on time} M = {Meets specifications}

P(T) = .75 P(M|T) = .80 P (M| not T ) = .60

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Probability Trees

P(T) = .75

P( ) = .25

T

P(M|T) = .80

P( |T) = .20

P(M| ) = .60

M

M

P(T and M) = .60

M

P( and M) = .15T

T

P( | ) = .40M

T

T

P(T and ) = .15

P( and ) = .10

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Probability Trees Find the probability that an order is on time and meets

specifications.

P(T and M) = .60

Find the probability that an order meets specifications.

P(M) = P(M and T) + P(M and not T)

= .60 + .15= .75

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Probability TreesExercise:

A manufacturer of snack crackers introduces several new

products each year. About 60% of the introductions are failures,30% are moderate successes, and 10% are major successes.

To try to improve the odds, the manufacturer tests new products

in a customer tasting panel. Of the failures, 50% receive a poor

rating in the panel, 30% a fair rating and 20% receive a good

rating, For the moderate successes, 20% receive a poor rating,40% a fair rating, and 40% a good rating. For major successes,

the percentages are 10% poor, 30% fair and 60% good.

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Probability Trees Find the joint probability of a new product being a failure and

receiving a poor rating.

Construct a probability table of all possible joint probabilities ofnew product results and panel rating.

If a new product receives a good rating, what is the probabilitythat the product will be a failure?

Create a probability tree using the given information. Use thetree to find the probability that a new product will be a majorsuccess, given that it gets a poor rating.

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Bayes Theorem Extension of Conditional Probability

Result generalizes to Mevents,A1,A2, ...,AM

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1

1

1 1

1 1 2 2

|

|| |

P A BP A B

P B

P B A P A

P B A P A P B A P A


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Bayes Theorem Importance: Update of prior beliefs

P(A1) priorbelief about probability ofA1

P(B|A1) probability relating information / signal to outcome(based on information about past relationship between signal andoutcome)

Bayes Theorem provides means to calculate posterior

probability P(A1 | B)

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Example: Market Research at

Arrow Cosmetics Arrow Cosmetics is considering the introduction of a

new location

Possible outcomes:

Units sold = {600K, 300K, 200K}

= {High, Medium, Low}

Subjective Probabilities:

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High 0.3

Medium 0.4

Low 0.3


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Additional Information Arrow can hire Marble Research to perform market

study test for new formula

Possible evaluations: {Hit, Flop}

Past experience with Marbles research:

P (Hit | High) = 1.0

P (Hit | Medium) = 0.375

P (Hit | Low) = 0.166

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Revising Probabilities Updating the a priori/marginal probability with additional

information (signal,test,symptom) to come up withrevised/posterior/conditional probability

Suppose that Marble declares that the new product willbe a hit

Should Arrow adjust its belief about the probability thatsales will be high? If so, how?

P(High) Test P(High | Hit)

P(Medium) Test P(Medium | Hit)

P(Low) Test P(Low | Hit)

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Lets Make a Deal! Suppose you are on a game show, and you are given the choice

of three doors. Behind one door is a car, behind the others,goats.

You pick a door, say number 1, and the host, who knows what'sbehind the doors, opens a door which has a goat. Say this isdoor 3.

The host asks you:

"Do you want to pick door number 2?"

Is it to your advantage to switch your choice of doors?

http://www.youtube.com/watch?v=mhlc7peGlGg

http://www.youtube.com/watch?v=P9WFKmLK0dc&feature=fvw

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http://www.youtube.com/watch?v=mhlc7peGlGghttp://www.youtube.com/watch?v=P9WFKmLK0dc&feature=fvwhttp://www.youtube.com/watch?v=P9WFKmLK0dc&feature=fvwhttp://www.youtube.com/watch?v=mhlc7peGlGg


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Summary Interpretations of probability (classical, frequency and subjective

approaches)

Rules and definitions of probability;The addition law for mutually exclusive events and non mutually

exclusive events, the complements law of probability.

Marginal, joint and conditional probabilities

Statistical Independence

Probability Trees

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Lecture 3 Topics:- Random variables

- Probability distributions, joint and marginal distributions

- Expected value, standard deviation (or variance) of a random

variable

- Covariance & Correlation

Things to do:- Go over these notes & Check Ch. 4.1, 4.2 for clarifications

- Read Chapter 4.3-4.9 in the textbook

- Attempt Questions 3-4 in Assignment 1

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