Assessment

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Lecture 8
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Assessment

• 70 of the grade will be based on assignments.
  These will be regular exercises similar to the
  two youve done.
• 30 of the grade will be the final.
• No mid-term.

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Frequency Theory

• Look at the number of actual outcomes, rather
  than possible outcomes. Probability is in the
  world.
• Finite frequentism the probability of an
  attribute A in a finite reference class B is the
  relative frequency of actual occurrences of A
  within B.
• So the probability of Heads is the number of
  actual Heads ever landed, divided by the number
  of flips.
• Probabilities are observable features of the
  world. So verifiable logical positivists.

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Reference classes

• According to frequentism, all probabilities are
  relativized to a reference class.
• Whats the probability that Bob, a 56 year old
  smoker, will have a heart attack within 12
  months?
• The relative frequency of heart attacks within 12
  months suffered by 56 year old smokers.
• Or perhaps the relative frequency of heart
  attacks within 12 months suffered by 56 year
  olds.
• Or perhaps the relative frequency of heart
  attacks within 12 months suffered by 56 year old
  smokers called Bob.
• Moral The probability depends on the reference
  class.

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Finite frequentism

• Admissible Yes.
• Ascertainable Not fully. We would have to know
  what happens on future flips.
• Applicable No

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The Problem of Single Cases

• What if the coin was only tossed once?
• Single cases Whats the probability of the Jets
  winning the Superbowl? 1 or 0?!
• There could never be a probability with an
  irrational number.
• Couldnt there be a freak run of all Heads? Would
  we really say the probabiltiy of Heads was more
  than 1/2?

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Counterfactual Frequentism

• The probability of Heads depends on the number of
  Heads there would have been if the coin had been
  repeatedly (infinitely?) tossed.
• Ascertainable No. How are we supposed to find
  this out? Not verifiable.
• General tension The more hypothetical the
  interpretation, the more difficult it is to find
  out the value. The less hypothetical, the less
  applicable it is i.e. tends to give us wrong
  answers.

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Propensity Interpretations

• How can we make sense of single cases?
• The probability is propensity / disposition /
  tendency of a certain experimental setup to
  produce a certain result.
• Mainly motivated by quantum mechanics.
• The frequencies are evidence for the probability,
  rather than being identical with the probability.

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• But what are propensities?
• There is some property that leads to this coin
  landing Heads on half the flips, but to call this
  property a propensity doesnt tell us anything
  about the property.
• Its like saying that a drug causes sleep in
  virtue of having a dormative virtue.

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Using ProbabilityOverview

• Proportions
• Distributions
• Correlations
• Axioms of probability
• Addition rules
• Multiplication rules
• Examples

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Proportions

• An urn contains some red marbles and some non-red
  marbles. (5.3)
• The number of red marbles divided by the total
  number of marbles is the proportion of red
  marbles.
• If there are 60 red marbles and 100 marbles, the
  proportion of red marbles is 60.

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Variables

• Marbles can be difference colours e.g. red, blue,
  black.
• Colour is a variable.
• Red, blue and black are values of the variable.

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Values are exclusive and exhaustive

• Exclusive Each marble can be only one colour at
  a time.
• In general, each member of the population can
  exhibit only one value.
• Example of non-exclusive properties Big and
  red.
• Exhaustive Each marble must have some colour.
• In general, each member of the population must
  exhibit some value.
• Example of exhaustive properties Red and not
  red.

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Distributions

• Proportions only tell us about one property.
• Distributions tell us about a range of
  properties. (5.4)
• The distribution is 20 blue, 30 green and 50
  red.

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Correlations

• A correlation is a relationship between two
  variables.
• So lets add the size of the marbles to the
  example.
• Variables Values
• Colour Red, Green
• Size Big, Small.

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• Is there correlation between two variables?
• Is there a correlation between size and colour?
• Is the proportion of large marbles among the red
  different from the proportion of large marbles
  among the green?
• If yes, correlation.
• If no, no correlation

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No Correlation

• 60 red, 40 green (5.7)
• Among the red, 45 large, 15 small
• Among the green, 30 large and 10 small
• The proportion of large marbles among the red is
  45/60 75
• The proportion of large marbles among the green
  is 30/40 75
• No correlation.

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Positive correlation

• 60 red, 40 green (5.8)
• Among the red, 45 large, 15 small
• Among the green, 10 large and 30 small
• The proportion of large marbles among the red is
  45/60 75
• The proportion of large marbles among the green
  is 10/40 25
• Positive correlation between large and red.

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• Age and height among children.
• Weight and nationality.
• Being technophobic and being a philosopher.
• Being a rock star and number of sexual partners.
• Taking a drug and getting a disease.
• Wearing white underwear and being born in summer.

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Negative correlation

• 60 red, 40 green (5.9)
• Among the red, 15 large, 45 small
• Among the green, 20 large and 40 small
• The proportion of large marbles among the red is
  15/60 25
• The proportion of large marbles among the green
  is 20/40 50
• Negative correlation between large and red.
• (Summary)

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Equivalences

• Negative correlation between large and red
• Positive correlation between large and green
• Negative correlation between small and green
• Positive correlation between small and red

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Symmetry How to Convert

• No correlation between large and red No
  correlation between small and red (5.10)
• 4530 75 large
• 1510 25 small
• How many are large and red? 45
• How many are small and red? 15
• 45/75 15/25 60, thus no correlation between
  small and red.

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Symmetry

• Positive correlation between large and red
  Positive correlation between red and large (5.11)
  
• 4510 55 large
• 1530 45 small
• 45 large and red
• 15 small and red
• 45 / 55 82
• 15 / 45 33
• Thus large is positively correlated with red.

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Measuring Correlation

• The strength of the correlation between being
  large and being red is the difference between the
  proportion of large marbles that are red and the
  proportion that are green.
• .75 - .25 0.5
• (But .82 .33 .49)

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Probability

• Let the probability of a value equal the
  proportion of objects with that value in the
  sample.
• A, B, C will be properties, or statements saying
  something about those properties i.e. a red ball
  will be selected.
• P(A), P(B)will be the probability of the
  property, or of the statement being true.

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Axioms of Probability

• 0. For all A, P(A) is between 0 and 1.
• 1. If A is a tautology, then P(A) 1
• 2. If A is a contradiction, then P(A) 0
• 3. If A and B are logically equivalent, then P(A)
  P(B)
• 4. If A and B are mutually exclusive, then P(A or
  B) P(A) P(B)

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Simple Addition Rule

• What is the probability that either a 1 or a 2
  will be rolled on a fair die?
• P(1) 1/6
• P(2) 1/6
• P(1 or 2) 1/6 1/6 2/6 1/3
• The probabilities can be added as long as the
  values are not exclusive. Why?

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The Senate
40
40
10
People not going to hell
Republicans
John McCain
P(Republican or going to hell) 0.4 0.4
(0.1) 0.7
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Addition Rules

• Simple addition rule
• If A and B are mutually exclusive, then P(A or
  B) P(A) P(B)
• General addition rule
• P(A or B) P(A) P(B) P(A and B)

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Multiplication rules

• If A and B are not correlated, then P(A and B)
  P(A) P(B).
• Example The probability of throwing two Heads in
  a row.
• P(Head on throw 1) ½
• P(Head on throw 2) ½
• P(Head on 1 and 2)
• P(Head on throw 1)P(Head on throw 2) ½ ½
  ¼

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• What if they are correlated? (5.8)
• Whats the probability that a red marble is
  large?
• P(Large) P(Red) .45 0.6
• But this is wrong.

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• Consider a streaky basketball shooter. He has a
  50 chance hitting his first shot. If he makes
  it, he has a 100 of hitting his second. If he
  doesnt, he has a 0 chance.
• What is the probability hell hit both?
• P(First) P(Second) 0.5 0.5 0.25
• But in fact the probability hell hit both is
  equal to the probabilty hell hit the first.

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Conditional Probability

• If Obama wins, whats the probability of a tax
  cut?
• Whats the probability of a tax cut given that
  Obama wins?
• P(Tax cut Obama wins)

Obama
Tax cut
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• Whats the probability that a red marble is
  large? (5.7)
• P(Large Red)
• Consider only the red marbles. What proportion is
  large?
• .75

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General Multiplication Rule

• P(Large and Red) P(Large Red) P(Red)
• 0.75 0.6
• P(A and B) P(AB) P(B)

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Assignment 3

• 5.1, 5.4 and 5.5.
• P.144