statistics & data analysis course numberb01.1305 course section31 meeting timewednesday...

Statistics & Data AnalysisCourse Number B01.1305

Course Section 31

Meeting Time Wednesday 6:00-8:50 pm

CLASS #2

Professor S. D. Balkin -- February 5, 2003

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Class #2 Outline

Brief review of last class Class introduction with Birthday Problem Questions on homework Chapter 3: A First Look at Probability


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Class Introduction and The Birthday Problem

Everyone introduce yourselves, giving your name, job/industry, and birthday

Question: How likely is it that two people in your class have the same birthday?

Let’s make a bet: I bet that at least two people in this class share the same birthday.• What should we bet?• Should I be so certain?


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Review of Last Class

Distinguish between quantitative and qualitative variables

Graphical representations of single variables

Numeric measures of center and variation

Chapter 3

A First Look At Probability


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Chapter Goals

Be able to interpret probabilities Understand the differences between statistics and probability Understand basic principles of probability

• Addition, Complements, Multiplication Understand statistical independence and conditional probability Be able to construct probability trees Understand managerial implications of probability


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Probability in Everyday Life

There is a 90% chance the Yankees will win the game tomorrow There is a sixty percent chance of thunderstorm this afternoon That bill has a 35% chance of being passed There is a 20% chance of rain today There is a 37% chance my hand will beat the dealer’s


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Probability in Everyday Life (cont)

Your company is deciding on launching a new product in the consumer market. Success based on reaction from competition, ability of suppliers to meet demand, unknown adverse events or issues, economic and regulatory conditions, etc.

An airplane has multiple engines and can make a journey safely as long as at least one is operating. Despite designers’ best efforts, what is the chance of a disaster occurring? Which parts of the plane should receive the most attention?

You’re still waiting for Ed McMahan to knock on your door?


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What is Probability?

Quantification of uncertainty and variability Basis for statistical inference and business decision making Probability theory is a branch of mathematics and it beyond

the scope of this class


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Illustrative Questions…

If you toss a coin, what is the probability of getting a head? If you toss a coin twice, what is the probability of getting exactly one

Head? • How can you verify your answer?

If you toss a coin 10 times and count the total number of Heads, do you think probability of 0 heads equals the probability of 5 heads?

Do you think probability of 4 heads equals the probability of 6 heads?


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History of Probability Originated from the study of games of chance

• Tossing a dice • Spinning a roulette wheel

Probability theory as a quantitative discipline arose in the seventeenth century when French gamblers prominent mathematicians for help in their gambling

In the eighteenth and nineteenth centuries, careful measurements in astronomy and surveying led to further advances in probability.

In the twentieth century probability is used to control the flow of traffic through a highway system, a telephone interchange, or a computer processor; find the genetic makeup of individuals or populations; figure out the energy states of subatomic particles; Estimate the spread of rumors; and predict the rate of return in risky investments.

Adapted from Probability Central

http://library.thinkquest.org/11506/


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Example: New York Times Online

Cellphones Not Killing Real Ones(May 26, 2002)

Despite their growing affection for cellphones, most Americans are not ready to pull the plug on traditional phones, according to a survey by

Maritz Research. The results were released this month.

When asked about the probability that they would use only cellphones for their calls in the next year, only 8 percent said that they were

very likely or certain to do so; 79 percent answered "very unlikely" or "absolutely not." Maritz surveyed 803 adults nationwide this spring.

Each respondent, or someone in the household, subscribed to a wireless phone service,

Forty-two percent, however, said their wireless phones had led them to use their existing long-distance companies less than they did

previously.

"Just five years ago, cellphones were viewed as a luxury; now they've become ingrained in everyday life for all members of a family," said

Paul Pacholski, a vice president at Maritz.


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Example: Wall Street Journal Online

European Markets Close Little Changed(May 21, 2002)

…Retail-price data published Tuesday showed that inflation in the United Kingdom was steady in April at an annual rate of 2.3%,

lower than the expected 2.4%. However, Lehman Brothers economist Michael Hume said the numbers are no obstacle to an

interest-rate hike. "We continue to look for a rate hike in June, but would put the probability of a move at no more than 60%,"

he said….


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Interesting Probability Quotes

Aristotle: The probable is what usually happens

Sir Arther Conan Doyle,The Sign of Four : When you have eliminated the impossible, what ever remains, however improbable, must be the truth.

Blaise Pascal: The excitement that a gambler feels when making a bet is equal to the amount he might win times the probability of winning it.


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Types of Occurrences

Predictable Occurrence: Occurrence whose value can be accurately determined using science:• Position of a meteor in 25 years

Unpredictable Occurrence: Occurrence whose value is based on a random process:• Toss of a coin

• Gender of a baby

Random Process: An event or phenomenon is called random if individual outcomes are uncertain but there is, however, a regular distribution of relative frequencies in a large number of repetitions.


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Probability and Statistics

Statistics: Observed data to generalizations about how the world works

Probability: Start from an assumption about how the world works, and then figure out what kinds of data you are likely to see

Probability is the only scientific basis for decision making in the

face of uncertainty


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Terminology

Random Experiment: A process or course of action that results in one of a number of possible outcomes• The outcome that occurs cannot be predicted with certainty

Outcome: Single possible results of a random experiment

Sample Space: The set of all possible outcomes of the experiment

Event: Any subset of the sample space

Simple Event: Event consisting of just one outcome


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Example

If we toss a nickel and a dime:• What are the possible outcomes?

• Which outcome is the event “no heads”?

• Which outcomes are in the event “one head and one tail”?

• Which outcomes are in the event “one or more heads”?


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Defining Probabilities

Probability has no precise definition!!

All attempts to define probability must ultimately rely on circular reasoning

Roughly speaking, the probability of an event is the chance or likelihood that the event will occur

To each event A, we want to attach a number P(A), called the probability of A, which represents the likelihood that A will occur


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Defining Probabilities (cont.)

There are various ways to define P(A), but in order to make sense, any definition must satisfy• P(A) is between zero and 1• P(E1) + P(E2) + ··· = 1, where E1, E2, ··· are the simple events

in the sample space

The three most useful approaches to obtaining a definition of probability are:• classical• relative frequency• subjective


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Classical Approach

Assume that all simple events are equally likely. Define the classical probability that an event A will occur as:

So P(A) is the number of ways in which A can occur, dividedby the number of possible individual outcomes, assuming allare equally likely.

S

AAP

in Events Simple #

in Events Simple #)(


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Example: Classical Approach

In tossing a coin twice, if we take:S = {HH, HT, TH, TT},then the classical approach assigns probability 1/ 4 to each simple event.

If A = {Exactly One Head} = {HT, TH}, thenP(A) = 2/ 4 = 1/ 2 .

Question : Does this tell you how often A would occur if we repeated the experiment (“toss a coin twice”) many times?


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Relative Frequency Approach

The probability of an event is the long run frequency of occurrence.

To estimate P(A) using the frequency approach, repeat the experiment n times (with n large) and compute x/n, where x = # Times A occurred in the n trials.

The larger we make n, the closer x/ n gets to P(A).

)(APn

x

Coin Flipping Example

http://shazam.econ.ubc.ca/flip/


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Classical and Frequency Approaches

If we can find a sample space in which the simple events really are equally likely, then the Law of Large Numbers asserts that the classical and frequency approaches will produce the same results.

For the experiment “Toss a coin once”, the sample space is S = {H, T} and the classical probability of Heads is 1/2.

According to the Law of Large Numbers (LLN), if we toss a fair coin repeatedly, then the proportion of Heads will get closer and closer to the Classical probability of 1/2.


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Subjective Approach

This approach is useful in betting situations and scenarios where one- time decision- making is necessary. In cases such as these, we wouldn’t be able to assume all outcomes are equally likely and we may not have any prior data to use in our choice.

The subjective probability of an event reflects our personal opinion about the likelihood of occurrence. Subjective probability may be based on a variety of factors including intuition, educated guesswork, and empirical data.

Eg: In my opinion, there is an 85% probability that Stern will move up in the rankings in the next Business Week survey of the top business schools.


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Example: Not Equally Likely Events

A market research survey asks the planned number of children for newly married couples giving the following data. What are the probabilities of a couple planning:

• 1 or 2 children?

• 3 or 4 children?

• 4 or more children?

Number of Children Probability

0 0.3471 0.2092 0.1913 0.1254 0.067

5+ 0.061


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Complement Rule

The probability of the complement of an event is equal to 1 minus the probability of the event itself

)(1)( APAP


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Example: Complement Rule

A market research survey asks the planned number of children for newly married couples giving the following data.

• Use the complement rule to find the probability of a couple planning to have any children at all

Number of Children Probability

0 0.3471 0.2092 0.1913 0.1254 0.067

5+ 0.061


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Odds

Odds are often used to describe the payoff for a bet. If the odds against a horse are a:b, then the bettor

must risk b dollars to make a profit of a dollars. If the true probability of the horse winning is b/(a+b),

then this is a fair bet. In the 1999 Belmont Stakes, the odds against Lemon

Drop Kid were 29.75 to 1, so a $2 ticket paid $61.50. The ticket returns two times the odds, plus the $2

ticket price.


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Example: Odds

If a fair coin is tossed once, the odds on Heads are 1 to 1

If a fair die is tossed once, the odds on a six are 5 to 1.

In the game of Craps, the odds on getting a 6 before a 7 are 6 to 5. (We will show this later).


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Combining Events

The union A B is the event consisting of all outcomes in A or in B or in both.

The intersection A B is the event consisting of all outcomes in both A and B.

If A B contains no outcomes then A, B are said to be mutually exclusive .

The Complement of the event A consists of all outcomes in the sample space S which are not in A.

A


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Combining Events (cont.)

occur NOT doesA y Probabilit)(

occurboth B andA y Probabilit)(

occurboth or Bor A y Probabilit)(

AP

BAP

BAP


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Rules for Combining Events

0)( : thenexclusive,mutually are B andA If

)()()()( :RuleAddition

)(1)( :Rule Complement

BAP

BAPBPAPBAP

APAP


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Example 1: Combining Events

Based on the past experience in your copier repair shop suppose… • Probability of a blown fuse is 6%• Probability of a broken wire is 4%• 1% of copiers to be repaired come in with both a blown fuse AND a

broken wire

What is the probability of a copier coming in with a blown fuse OR a broken wire?


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Example 2: Combining Events

Market research firm tests a potential new product 200 male respondents, selected at random, gave their

opinions for the product and their marital status giving the following data:

Poor Fair Good Excellent Total

Never Married 5 9 26 10 50Divorced 1 4 16 9 30Married 12 23 37 32 104Widowed 2 8 5 1 16Total 20 44 84 52 200

Opinion

Marital Status


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Conditional Probability

Calculating probabilities given some restrictive condition Example: Absenteeism Last Year for 400 Employees.

Compute the probability that a randomly selected employee is a smoker.

If we are told that the employee was absent less than 10 days, does this partial knowledge change the probability that the employee is a smoker?

Days Absent Smoker Non-Smoker Total

<10 34 260 29410+ 78 28 106Total 112 288 400


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Conditional Probability (cont.)

)(

)()|(

)|(y Probabilit lConditiona

AP

BAPABP

ABP

304.00.28

0.085 days) 10 Absent |Smoker (

P


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Multiplication Law

)|()(

)|()()(

B, andA eventsany For

BAPBP

ABPAPBAP


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Statistical Independence

Events A and B are statistically independent if and only if P(B|A) = P(B). Otherwise, they are dependent.

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


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Example: Independence

Seattle corporations with 500 or more employees• 468 executives; 30 whom are women• Conditional probability of a person being a woman given that the

person is an executive is 30/468 = 0.064

In the population, 51.2% are women

Since the probability of randomly choosing a women changes when conditioning on “being an executive”, being a women and being an executive are dependent events


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Another Independence Example You are responsible for scheduling a construction project

• In order to avoid trouble, it will be necessary for the foundation to be completed by July 27th and for the electricity to be installed before August 6th

• Based on your experiences, you fix probabilities of 0.83 and 0.91 for these events to occur

• Assume you have a 96% chance of meeting one deadline or the other (or both)

What is the probability of missing both deadlines?

Are these events mutually exclusive? How?

Are these events independent? How?


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Revisiting the Birthday Problem

What is the probability that at least two people in this class share the same birthday?

Can be formulated as: What is the probability no one in this class shares the same birthday, and take the complement


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Probability Tables and Trees

Human resources found that 46% of its junior executives have two-career marriages, 37% have single-career marriages, and 17% are unmarried.

HR estimates that 40% of the two-career marriage executives would refuse to transfer, as would 15% of the single-career-marriage executives, and 10% of the unmarried executives.

If a transfer offer is made to randomly selected executives, what is the probability it will be refused?


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Probability Tables

Fill in this probability table:

Two-Career Single-Career Unmarried

Refused

Accepted0.46 0.37 0.17


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Constructing Probability Trees

1. Events forming the first set of branches must have known marginal probabilities, must be mutually exclusive, and should exhaust all possibilities

2. Events forming the second set of branches must be entered at the tip of each of the sets of first branches. Conditional probabilities, given the relevant first branch, must be entered, unless assumed independence allows the use of unconditional probabilities

3. Branches must always be mutually exclusive and exhaustive


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Probability Tree

Construct a probability tree


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Let’s Make a Deal

In the show Let’s Make a Deal, a prize is hidden behind on of three doors. The contestant picks one of the doors.

Before opening it, one of the other two doors is opened and it is shown that the prize isn’t behind that door.

The contestant is offered the chance to switch to the remaining door.

Should the contestant switch? Solve by making a tree…


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Employee Drug Testing

A firm has a mandatory, random drug testing policy

The testing procedure is not perfect.• If an employee uses drugs, the test will be positive with probability

0.90.• If an employee does not use drugs, the test will be negative 95% of the

time.• Confidential sources say that 8% of the employees are drug users

8% is an unconditional probability; 90 and 95% are conditional probabilities


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Employee Drug Testing (cont.)

Create a probability tree and verify the following probabilities:• Probability of randomly selecting a drug user who tests positive =

0.072• Probability of randomly selecting a non-user who tests positive = 0.046• Probability of randomly selecting someone who tests positive = 0.118• Conditional probability of testing positive given a non-drug user = 0.05


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Next Time…

Random variables and probability distributions


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Homework #2

Hildebrand/Ott• 3.3• 3.4• 3.5• 3.8• 3.10, 3.11, and 3.12 on pages 76-77. These all draw on

the same data, so it’s easy to deal with them together. Note that those who recalled the commercial correctly are in the “favorable” and “unfavorable” columns.

• 3.14• 3.24, pages 90-91.

Observe that the rows of the given table sum to 1. These are thus conditional probabilities for the retest, given the results of the first test. For example, P(Retest = minor | First = major) = 0.5. Part (c) asks you to supply two numbers.

• 3.28• 3.29

Verzani• NONE

statistics & data analysis course numberb01.1305 course section31 meeting timewednesday...

Documents

class slide

probability trees

probability theory

conditional probability

history of probability

class class introduction

variation slide

pm class