Today

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Today

• Today Course Outline, Start Chapter 1
• Assignment 1
• Read Chapter 1 by next Tuesday

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Statistics

• What is Statistics?
• Discipline that deals with the collection,
  summary, organization and interpretation of data
• Used to help predict and answer questions about
  real processes

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Example

• Dr. B. Spock was convicted of conspiracy during
  Vietnam era
• Convicted by an all-male jury
• Felt trial was unfair because no women were on
  the jury
• Jury list for Spocks judge had 14.4 women
• Jury list for 6 other judges had 29 women
• Does this seem fair?

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Example

• Questions
• What is the chance or probability that in a
  population with 29 women, we could select a jury
  pool of only 14.4 women?
• Is 14.4 close to 29?
• What does chance or probability mean?

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Sample Space

• The term experiment is used to denote an activity
  which produces an unpredictable outcome
• Will describe and summarize experiments using
  models
• Model describing an experiment should to take
  into account all possible outcomes
• Collection of all possible outcomes of an
  experiment is called the sample space and is
  denoted by

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• Example
• A balanced coin is flipped
• Do not know outcome in advance
• What if we tossed the coin 2 times?
• Example
• Roll of a fair die
• Example
• Number of people arriving in an emergency room in
  before one dies

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Event

• A collection of outcomes (a subset of the sample
  space) is called an event
• An event, E, is said to occur if one of its
  outcomes occurs

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Event

• Example
• A balanced coin is flipped two times
• Let E be the event that a tails occurs
• E
• Example
• Committee of two people is chosen from 5 people
  (a-e)
• Let E be the event that person a is on the
  committee
• E

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Event Algebra

• We will generally introduce these ideas as needed

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Event Algebra

• Example Fair Die
• E1E3

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Venn Diagrams

• Can represent sample space and events using a
  Venn Diagram

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Probability for Experiments

• A probability model assigns probabilities to
  outcomes and/or events in the sample space
• Simplest case is when the number of possible
  outcomes in the sample space is finite and each
  outcome is equally likely
• If there are N outcomes in the sample space and
  each outcome is equally likely then the
  probability of any individual outcome is 1/N
• Example Fair die is rolled. What is the
  probability of observing 6?

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Probability for Experiments

• An event E is a collection of outcomes from the
  sample space
• Recall, an event, E, is said to occur if one of
  its outcomes occurs
• So, P(E)

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Probability for Experiments

• Example
• In roulette, the possible outcomes are
  00,0,1,2,36
• Outcomes 00 and 0 are green, odd outcomes are red
  and even outcomes are black
• Let R be the event that a red is observed in one
  spin of the roullette wheel
• P(R)
• Example
• In the committees example, the probability that
  person a is on the committee is

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Properties of Probability

• These are called the axioms of probability

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Consequences of the Axioms

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• Example
• Small university offers 2 language courses,
  French and German
• Everyone is required to take a language course
• If 75 of students take French and 55 take
  German what proportion of students take both
  courses?
• What proportion take only one course?

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Combinatorics

• In the equally likely case, computing
  probabilities involves counting
• This can be hardreally
• Combinatorics is a branch of mathematics whichs
  develops efficient counting methods

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Combinatorics

• Consider the rhyme
• As I was going to St. Ives
• I met a man with seven wives
• Every wife had seven sacks
• Every sack had seven cats
• Every cat had seven kits
• Kits, cats, sacks and wives
• How many were going to St. Ives?
• Answer

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Combinatorics