Stat 155, Section 2, Last Time

1
Stat 155, Section 2, Last Time

• Producing Data How to Sample?
• Placebos
• Double Blind Experiment
• Random Sampling
• Statistical Inference
• Population parameters , ,
• Sample statistics , ,
• (keep these separate)
• Probability Theory

2
Reading In Textbook

• Approximate Reading for Todays Material
• Pages 231-240, 256-257
• Approximate Reading for Next Class
• Pages 259-271, 277-286

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Chapter 4 Probability

• Goal quantify (get numerical) uncertainty
• Key to answering questions above
• (e.g. what is natural variation
• in a random sample?)
• (e.g. which effects are significant)
• Idea Represent how likely something is by a
  number

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Probability

• Recall Basics
• Assign numbers (representing how likely),
• to outcomes
• E.g. Die Rolling
• Pcomes up 4 1/6
• Outcome is 4
• Probability is 1/6

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Simple Probability

• Quantify how likely outcomes are by assigning
  probabilities
• I.e. a number between 0 and 1, to each outcome,
  reflecting how likely
• Intuition
• 0 means cant happen
• ½ means happens half the time
• 1 means must happen

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Simple Probability

• Main Rule
• Sum of all probabilities (i.e. over all
  outcomes) is 1
• E.g. for die rolling

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Simple Probability

• HW
• 4.13a
• 4.15

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Probability

• General Rules for assigning probabilities
• Frequentist View
• (what happens in many repititions?)
• Equally Likely for n outcomes
• Pone outcome 1/n (e.g. die rolling)
• iii. Based on Observed Frequencies
• e.g. life tables summarize when people die
• Gives prob of dying at a given age
• life expectancy

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Probability

• General Rules for assigning probabilities
• Personal Choice
• Reflecting your assessment
• E.g. Oddsmakers
• Careful requires some care
• (key is probs need to sum to 1)
• HW
• 4.19

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Probability - Events

• More Terminology (to carry this further)
• An event is a set of outcomes
• Die Rolling an even , is the event 2, 4, 6
• Notes
• If betting on even dont care about , only even
  or odd
• Thus events are our foundation
• Each outcome is an event the set containing
  just that outcome
• So event is the more general concept

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Probability on Events

• Sample Space is the set of all outcomes
• event with everything that can happen
• Extend Probability to Events by
• Pevent sum of probs of outcomes in event

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Probability

• Technical Summary
• A probability model is a sample space
• I.e. set of outcomes, plus a probability, P
• P assigns numbers to events,
• Events are sets of outcomes

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Probability Function

• The probability, P, is a function,
• defined on a set of events
• Recall function in math
• plug-in
  get out
• Probability Pevent how likely

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Probability Function

• E.g. Die Rolling
• Sample Space 1, 2, 3, 4, 5, 6
• an even is the event 2, 4, 6 (a set)
• Peven P2, 4, 6
• P2 P4 P6
• 1/6 1/6 1/6 3/6 ½
• Fits, since expect even half the time

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Probability HW

• HW
• 4.11
• 4.13b
• 4.17

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And now for something completely different

• Did you here about the constipated mathematician?

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And now for something completely different

• Did you here about the constipated mathematician?
• He worked it out with a pencil!

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And now for something completely different

• Did you here about the constipated mathematician?
• He worked it out with a pencil!
• Apologies for the juvenile nature

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And now for something completely different

• Did you here about the constipated mathematician?
• He worked it out with a pencil!
• Apologies for the juvenile nature
• But there is an important point

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And now for something completely different

• Did you here about the constipated mathematician?
• He worked it out with a pencil!
• Apologies for the juvenile nature
• But there is an important point
• The pencil is a powerful
• mathematical tool

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And now for something completely different

• The pencil is a powerful
• mathematical tool
• An old student
• I was once good in math
• But suddenly lost that
• Reason tried to do too much in head
• Reason never learned power of the pencil

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And now for something completely different

• The pencil is a powerful
• mathematical tool
• For us now is time to start using pencil
• I do PowerPoint in class
• You use pencil on HW (and exams)
• Change in mindset, from Excel

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Probability

• Now stretch ideas with more interesting e.g.
• E.g. Political Polls, Simple Random Sampling
• 2 views
• Each individual equally likely to be in sample
• Each possible sample is equally likely
• Allows for simple Probability Modelling

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Simple Random Sampling

• Sample Space is set of all possible samples
• An Event is a set of some samples
• E.g. For population A, B, C, D
• Each is a voter
• Only 4, so easy to work out

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S. R. S. Example

• For population A, B, C, D,
• Draw a S. R. S. of size 2
• Sample Space
• (A,B), (A,C), (A,D), (B,C), (B,D), (C,D)
• outcomes, i.e. possible samples of size 2

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S. R. S. Example

• Now assign P, using equally likely rule
• PA,B PA,C PC,D
• 1/(samples) 1/6
• An interesting event is
• C in sample (A,C),(B,C),(D,C)
• (set of samples with C in them)

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S. R. S. Example

• PC in sample
• i.e. happens half the time.

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S. R. S. Probability HW

• HW C10
• Abby, Bob, Mei-Ling, Sally and Roberto work for a
  firm. Two will be chosen at random to attend an
  overseas meeting. The choice will be made by
  drawing names from a hat (this is an S. R. S. of
  2).
• Write down all possible choices of 2 of the 5
  names. This is the sample space.
• Random choice makes all choices equally likely.
  What is the probability of each choice? (1/10)
• What is the prob. that Sally is chosen? (4/10)
• What is the prob. that neither Bob, nor Roberto
  is chosen? (3/10)

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Political Polls Example

• What is your chance of being in a poll of 1000,
  from S.R.S. out of 200,000,000?
• (crude estimate of of U. S. voters)
• Recall each sample is equally likely so
• Problem this is really big
• (5,733 digits, too big for easy handling.)

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Political Polls Example

• More careful calculation
• Makes sense, since you are equally likely to be
  in samples

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And now for something completely different

• .

An interesting phone conversation. Sound File
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Probability

• Now have prob. models
• But still hard to work with
• E.g. probs we care about, such as accuracy
  estimators, need better tools
• Need to look more deeply

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3 Big Rules of Probability

• Main idea calculate complicated probs
• By decomposing events in terms of simple events
• Then calculating probs of these
• And then using simple rules of probabilty to
  combine

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3 Big Rules of Probability

• Rule I the not rule
• Pnot A 1 PA
• Why?
• E.g. equally likely sample points
• And more generally

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The Not Rule of Probability

• Text Book Terminology (sec. 4.2)
• not A
• for complement
• (set theoretic term)
• (I prefer not, since more intuitive)

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The Not Rule of Probability

• HW Rework, using the not rule
• 4.17b

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3 Big Rules of Probability

• Rule II the or rule
• PA or B PA PB PA and B
• Why?
• E.g. equally likely sample points
• Helpful Pic

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Big Rules of Probability

• E.g. Roll a die,
• Let A 4 or less 1, 2, 3, 4
• Let B Odd 1, 3, 5
• Check how rules work by calculating 2 ways
• Direct Pnot A P5, 6 2/6 1/3
• By Rule I Pnot A 1 PA 1 4/6 1/3

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The Or Rule of Probability

• A 4 or less 1, 2, 3, 4
• B Odd 1, 3, 5
• Check how rule works by calculating 2 ways
• Direct PA or B P1, 2, 3, 4, 5 5/6
• By Rule II PA or B
• PA PB PA and B
• 4/6 3/6 2/6 5/6 (check!)

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The Or Rule of Probability

• Seems too easy?
• Dont really need rules for these simple things
• But they are the key to bigger problems
• Such as Simple Random Sampling
• HW 4.86 (0.317)

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The Or Rule of Probability

• E.g A college has 60 Women and 40 smokers,
  and 50 women who dont smoke.
• What is the chance that a randomly selected
  student is either a women or a non-smoker?
• (seems gt60, but twice? Must be lt 100, i.e.
  must be some overlap)

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College Women Smokers E.g.

• PW or NS PW PNS PW NS
• (choice of letters make easy to work with)
• 0.6 (1 0.4) 0.5 0.7
• (answer is 70 Women or Non-Smokers)
• Note rules are powerful when used together

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More Or Rule HW

• HW C11
• A building company bids on two large projects.
  The president believes the chance of winning the
  1st is 0.6, the chance of winning the 2nd is 0.5,
  and the chance of winning both is 0.3. What is
  the chance of winning at least one of the jobs?
  (0.8)

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The Or Rule of Probability

• E.g. Events A B are mutually exclusive, i.e.
  disjoint, when PA B 0
• (i.e. no chance of seeing both at same time)
• Useful Pic
• Then
• PA or B PA PB
• Text suggests new rule, I say special case

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The Exclusive Or Rule

• HW C12
• Choose an acre of land in Canada at random. The
  probability is 0.35 that it is forest, and 0.03
  that it is pasture.
• What is the probability that the acre chosen is
  not forested? (0.65)
• What is the probability that it is either forest
  or pasture? (0.38)
• What is the probability that a randomly chosen
  acre in Canada is neither forest nor pasture?
  (0.62)