Today

1
Today

• Today Some more counting examples Start
  Chapter 2
• Important Sections from Chapter 1 1.1-1.5
  Please read 1.7-1.8
• Reading
• Assignment 2 is up on the web site
• www.stat.lsa.umich.edu/dbingham/stat405
• Please read Chapter 2
• Suggested problems 2.4, 2.5, 2.7, 2.13, 2.25,
  2.28, 2.32, 2R1, 2R2

2
Example

• A batch of 50 manufactured items contains 5
  defective items. If 10 items are randomly
  selected what is the probability that at least
  one of the defectives is found?

3
Example

• If there are 60 people in a Statistics 405 class,
  what is the probability that at least 2 have the
  same birthday?

4
Partitions

• A partition of an event F is a collection of
  events (E1, E2, , Ek) that are mutually disjoint
  (i.e., EiEj , for all i and j ) and where
• Since none of the Eis have outcomes in common,
  then the number of outcomes in F is the same as
  the sum of the outcomes in each of the Eis
• That is, if F has N outcomes and Ei has Ni
  outcomes, then NN1N2Nk

5
Partitions

• If E1, E2,, Ek form a partition of F, then the
  number of distinguishable ways to partition N
  objects into k groups is

6
Example

• What is the number of ways that 52 cards may be
  divided equally among 4 players

7
Chapter 2

• Chapter 2 has two main topics
• Random variables and their distributions
• Conditional probability and independence
• Will begin with the second topic

8
Conditional Probability

• Example
• A family has two children
• The sample space for the possible gender of their
  children is gg, gb, bg, bb.
• The parents are seen leaving a Girl Scout
  meeting, suggesting that at least one of their
  children is a girl
• If each gender is equally likely, what is the
  probability that the other child is a boy

9
Conditional Probability

• Consider a probability model with sample space
• Let E be an event in the sample space
• If F is another event in the sample space where
  P(F)gt0, the conditional probability of E given F
  is
• Idea This represents the appropriate change in
  the probability assignment when we know that F
  has occurred

10
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?

11
Some Useful Formulas

• Multiplication Rule
• Law of Total Probability
• Bayes Theorem

12
Where do these come from?
13
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

14
Several Events

• Suppose (E1, E2, , Ek) form a partition of the
  sample space
• Bayes Theorem suppose (E1, E2, , Ek) form a
  partition of the sample space

15
Example (2.28 from text)

• Evidence in a paternity suit indicates that 4
  particular men are equally likely to be the
  father of the child
• Both the child and the mother have type O blood
• The table below give the blood type and the
  probability of producing a type O offspring with
  a type O mother
• What is the probability that man 1 is the father?
  Man 3? Man 3?

16
Independence

• For a given probability model and events E and F,
  the two events are said to be independent iff
  P(EF)P(E)P(F)
• Another way of viewing this is that for events E
  and F, the two events are said to be independent
  iff P(EF)P(E)

17
Example

• A single card is drawn from a standard deck
• AAce
• BSpade
• Are these events independent?

18
Example

• Consider a pair of genes which may be one of two
  alleles a or A
• This could represent a smooth (a) or wrinkled
  seed (A)
• According to Medelain Theory, each parent
  contributes either a or A to the offspring
  independently
• E1parent 1 gives a
• E2parent 2 gives a
• P(aa)