Probability, Bayes Theorem and the Monty Hall Problem

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Probability, Bayes Theorem and the Monty Hall
Problem
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Probability Distributions

• A random variable is a variable whose value is
  uncertain.
• For example, the height of a randomly selected
  person in this class is a random variable I
  wont know its value until the person is
  selected.
• Note that we are not completely uncertain about
  most random variables.
• For example, we know that height will probably be
  in the 5-6 range.
• In addition, 56 is more likely than 50 or
  60 (for women).
• The function that describes the probability of
  each possible value of the random variable is
  called a probability distribution.

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Probability Distributions

• Probability distributions are closely related to
  frequency distributions.

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Probability Distributions

• Dividing each frequency by the total number of
  scores and multiplying by 100 yields a percentage
  distribution.

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Probability Distributions

• Dividing each frequency by the total number of
  scores yields a probability distribution.

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Probability Distributions

• For a discrete distribution, the probabilities
  over all possible values of the random variable
  must sum to 1.

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Probability Distributions

• For a discrete distribution, we can talk about
  the probability of a particular score occurring,
  e.g., p(Province Ontario) 0.36.
• We can also talk about the probability of any one
  of a subset of scores occurring, e.g., p(Province
  Ontario or Quebec) 0.50.
• In general, we refer to these occurrences as
  events.

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Probability Distributions

• For a continuous distribution, the probabilities
  over all possible values of the random variable
  must integrate to 1 (i.e., the area under the
  curve must be 1).
• Note that the height of a continuous distribution
  can exceed 1!

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Continuous Distributions

• For continuous distributions, it does not make
  sense to talk about the probability of an exact
  score.
• e.g., what is the probability that your height is
  exactly 65.485948467 inches?

Normal Approximation to probability distribution
for height of Canadian females (parameters from
General Social Survey, 1991)
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Continuous Distributions

• It does make sense to talk about the probability
  of observing a score that falls within a certain
  range
• e.g., what is the probability that you are
  between 53 and 57?
• e.g., what is the probability that you are less
  than 510?

Normal Approximation to probability distribution
for height of Canadian females (parameters from
General Social Survey, 1991)
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Probability of Combined Events
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Probability of Combined Events
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Exhaustive Events

• Two or more events are said to be exhaustive if
  at least one of them must occur.
• For example, if A is the event that the
  respondent sleeps less than 6 hours per night and
  B is the event that the respondent sleeps at
  least 6 hours per night, then A and B are
  exhaustive.
• (Although A is probably the more exhausted!!)

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Independence
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An Example The Monty Hall Problem
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Problem History

• When problem first appeared in Parade,
  approximately 10,000 readers, including 1,000
  PhDs, wrote claiming the solution was wrong.
• In a study of 228 subjects, only 13 chose to
  switch.

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Intuition

• Before Monty opens any doors, there is a 1/3
  probability that the car lies behind the door you
  selected (Door 1), and a 2/3 probability it lies
  behind one of the other two doors.
• Thus with 2/3 probability, Monty will be forced
  to open a specific door (e.g., the car lies
  behind Door 2, so Monty must open Door 3).
• This concentrates all of the 2/3 probability in
  the remaining door (e.g., Door 2).

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Analysis
Car hidden behind Door 3
Car hidden behind Door 2
Car hidden behind Door 1
Player initially picks Door 1
Host must open Door 2
Host must open Door 3
Host opens either Door 2 or 3
Switching loses with probability 1/6
Switching wins with probability 1/3
Switching wins with probability 1/3
Switching loses with probability 1/6
Switching wins with probability 2/3
Switching loses with probability 1/3
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Notes

• It is important that
• Monty must open a door that reveals a goat
• Monty cannot open the door you selected
• These rules mean that your choice may constrain
  what Monty does.
• If you initially selected a door concealing a
  goat, then there is only one door Monty can open.
• One can rigorously account for the Monty Hall
  problem using a Bayesian analysis

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End of Lecture 2

• Sept 17, 2008

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Conditional Probability

• To understand Bayesian inference, we first need
  to understand the concept of conditional
  probability.
• What is the probability I will roll a 12 with a
  pair of (fair) dice?
• What if I first roll one die and get a 6? What
  now is the probability that when I roll the
  second die they will sum to 12?

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Conditional Probability

• The conditional probability of A given B is the
  joint probability of A and B, divided by the
  marginal probability of B.
• Thus if A and B are statistically independent,
• However, if A and B are statistically dependent,
  then

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Bayes Theorem

• Bayes Theorem is simply a consequence of the
  definition of conditional probabilities

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Bayes Theorem

• Bayes theorem is most commonly used to estimate
  the state of a hidden, causal variable H based on
  the measured state of an observable variable D

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Bayesian Inference

• Whereas the posterior p(HD) is often difficult
  to estimate directly, reasonable models of the
  likelihood p(DH) can often be formed. This is
  typically because H is causal on D.
• Thus Bayes theorem provides a means for
  estimating the posterior probability of the
  causal variable H based on observations D.

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Marginalizing

• To calculate the evidence p(D) in Bayes
  equation, we typically have to marginalize over
  all possible states of the causal variable H.

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The Full Monty

• Lets get back to The Monty Hall Problem.
• Lets assume you initially select Door 1.
• Suppose that Monty then opens Door 2 to reveal a
  goat.
• We want to calculate the posterior probability
  that a car lies behind Door 1 after Monty has
  provided these new data.

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The Full Monty
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The Full Monty
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But were not on Lets Make a Deal!

• Why is the Monty Hall Problem Interesting?
• It reveals limitations in human cognitive
  processing of uncertainty
• It provides a good illustration of many concepts
  of probability
• It get us to think more carefully about how we
  deal with and express uncertainty as scientists.
• What else is Bayes theorem good for?

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Clinical Example

• Christiansen et al (2000) studied the mammogram
  results of 2,227 women at health centers of
  Harvard Pilgrim Health Care, a large HMO in the
  Boston metropolitan area.
• The women received a total of 9,747 mammograms
  over 10 years. Their ages ranged from 40 to 80.
  Ninety-three different radiologists read the
  mammograms, and overall they diagnosed 634
  mammograms as suspicious that turned out to be
  false positives.
• This is a false positive rate of 6.5.
• The false negative rate has been estimated at 10.

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Clinical Example

• There are about 58,500,000 women between the ages
  of 40 and 80 in the US
• The incidence of breast cancer in the US is about
  184,200 per year, i.e., roughly 1 in 318.

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Clinical Example