Statistics

1
Statistics Data Analysis

• Course Number B01.1305
• Course Section 31
• Meeting Time Wednesday 600-850 pm

CLASS 2
2
Class 2 Outline

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

3
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?
• Lets 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?

4
Review of Last Class

• Distinguish between quantitative and qualitative
  variables
• Graphical representations of single variables
• Numeric measures of center and variation

5
Chapter 3
A First Look At Probability
6
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

7
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
  dealers

8
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?
• Youre still waiting for Ed McMahan to knock on
  your door?

9
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

10
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?

11
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

12
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.

13
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.

14
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.

15
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.

16
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

17
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

18
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?

19
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

20
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

21
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, divided
• by the number of possible individual outcomes,
  assuming all
• are equally likely.

22
Example Classical Approach

• In tossing a coin twice, if we takeS 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?

23
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).

Coin Flipping Example
24
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.

25
Subjective Approach

• This approach is useful in betting situations and
  scenarios where one- time decision- making is
  necessary. In cases such as these, we wouldnt 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.

26
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?

27
Complement Rule

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

28
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

29
Odds

• Odds are often used to describe the payoff for a
  bet.
• If the odds against a horse are ab, then the
  bettor must risk b dollars to make a profit of a
  dollars.
• If the true probability of the horse winning is
  b/(ab), 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.

30
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).

31
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.

32
Combining Events (cont.)
33
Rules for Combining Events
34
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?

35
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

36
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?

37
Conditional Probability (cont.)
38
Multiplication Law
39
Statistical Independence

• Events A and B are statistically independent if
  and only if P(BA) P(B). Otherwise, they are
  dependent.
• If events A and B are independent, then P(A ? B)
  P(A)P(B)

40
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

41
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?

42
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

43
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?

44
Probability Tables

• Fill in this probability table

45
Constructing Probability Trees

• Events forming the first set of branches must
  have known marginal probabilities, must be
  mutually exclusive, and should exhaust all
  possibilities
• 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
• Branches must always be mutually exclusive and
  exhaustive

46
Probability Tree

• Construct a probability tree

47
Lets Make a Deal

• In the show Lets 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 isnt
  behind that door.
• The contestant is offered the chance to switch to
  the remaining door.
• Should the contestant switch?
• Solve by making a tree

48
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

49
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

50
Next Time

• Random variables and probability distributions

51
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 its 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