Statistics 270 - Lecture 6 - PowerPoint PPT Presentation

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Title: Statistics 270 - Lecture 6


1
Statistics 270 - Lecture 6
2
  • Last day Probability rules
  • Today Conditional probability
  • Suggested problems Chapter 2 45, 47, 59, 63, 65

3
  • Example Lets Make a Deal
  • A contestant is given a choice of three doors of
    which one contained a prize such as a Car
  • The other two doors contained gag gifts like a
    chicken or a donkey
  • After the contestant choses an initial door, the
    host of the show reveals an empty door among the
    two unchosen doors, and asks the contestant if
    they would like to switch to the other unchosen
    door
  • What should the contestant do?

4
Conditional Probability
  • Sometimes interested in in probability of an
    event, after information regarding another event
    has been observed
  • The conditional probability of an event A, given
    that it is known B has occurred is
  • Called probability of A given B

5
  • Example Lets Make a Deal
  • A contestant is given a choice of three doors of
    which one contained a prize such as a Car
  • The other two doors contained gag gifts like a
    chicken or a donkey
  • After the contestant choses an initial door, the
    host of the show reveals an empty door among the
    two unchosen doors, and asks the contestant if
    they would like to switch to the other unchosen
    door
  • What should the contestant do?

6
Some Useful Formulas
  • Multiplication Rule
  • Law of Total Probability
  • Bayes Theorem

7
Where do these come from?
8
Example
  • In a region 12 of adults are smokers, 0.8 are
    smokers with emphysema and 0.2 are non-smokers
    with emphysema
  • What is the probability that a randomly selected
    individual has emphysema?
  • Given that the person is a smoker, what is the
    probability that the person has emphysema?

9
Example
  • From a group of 5 Democrats, 5 Republicans and 5
    Independents, a committee of size 3 is to
    selected
  • What is the probability that each group will be
    represented on the committee if the first person
    selected is an Independent?

10
Example
  • Consider a routine diagnostic test for a rare
    disease
  • Suppose that 0.1 of the population has the
    disease, and that when the disease is present the
    probability that the test indicates the disease
    is present is 0.99
  • Further suppose that when the disease is not
    present, the probability that the test indicates
    the disease is present is 0.10
  • For the people who test positive, what is the
    probability they actually have the disease

11
Example (Randomized Response Model)
  • Can design survey using conditional probability
    to help get honest answer for sensitive questions
  • Want to estimate the probability someone cheats
    on taxes
  • Questionnaire
  • 1. Do you cheat on your taxes?
  • 2. Is the second hand on the clock between 12 and
    3?
  • YES NO
  • Methodology Sit alone, flip a coin and if the
    outcome is heads answer question 1 otherwise
    answer question 2

12
Several Events
  • Suppose (A1, A2, , Ak) form a partition of the
    sample spacei.e., they are mutually exclusive
    and their union equals the sample space
  • Bayes Theorem suppose (A1, A2, , Ak) form a
    partition of the sample space

13
Independent Events
  • Two events are independent if
  • The intuitive meaning is that the outcome of
    event B does not impact the probability of any
    outcome of event A
  • Alternate form

14
Example
  • Flip a coin two times
  • S
  • Ahead observed on first toss
  • Bhead observed on second toss
  • Are A and B independent?

15
Example
  • Mendel used garden peas in experiments that
    showed inheritance occurs randomly
  • Seed color can be green or yellow
  • G,GGreen otherwise pea is yellow
  • Suppose each parent carries both the G and Y
    genes
  • M Male contributes G F Female contributes
    G
  • Are M and F independent?

16
Several Independent Events
  • Events A1, A2, , An are mutually independent if
    for every k (k2, 3, , n) and every index set
    i1, , ik
  • That is, events are mutually independent if the
    probability of the intersection of any subset of
    the n events is equal to the product of the
    individual probabilities
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