Probability

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Probability

• Scott Matthews
• Courses 12-706 / 19-702/ 73-359
• Lecture 15 - 10/19/2005

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Admin Issues

• HW 4, Project 1 due today
• Lecture
• Wednesday First Project Ideas

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Conditional Probability

• Probability (P) that A occurs conditional on B
  occurring
• Also referred to as P of A given B
• Joint Probability P(A and B)
• Recall Venn diagram from chapter - finding
  portion of Dow Jones Up circle where Stock
  Price Up occurs.

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Total Probability

• Alternatively..
• Probability of an event occuring alone is
  combination of all possible joint outcomes with
  another event
• Given n mutually exclusive events (A1..An) whose
  probabilities sum to 1

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Another Conditional Example

• Example of Probability of having passed HW 1 and
  HW 2 (assume pass75).

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Subjective Probabilities (Chap 8)

• Main Idea We all have to make personal
  judgments (and decisions) in the face of
  uncertainty (Granger Morgans career)
• These personal judgments are subjective
• Subjective judgments of uncertainty can be made
  in terms of probability
• Examples My house will not be destroyed by a
  hurricane. The Pirates will have a winning
  record (ever). Driving after I have 2 drinks
  is safe.

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Outcomes and Events

• Event something about which we are uncertain
• Outcome result of uncertain event
• Subjectively once event (eg coin flip) has
  occurred, what is our judgment on outcome?
• Represents degree of belief of outcome
• Long-run frequencies, etc. irrelevant
• Example Steelers play AFC championship game at
  home. I Tivo it instead of watching live. I
  assume before watching that they will lose.
• Insert Cubs, Astros, etc. as needed (Sox removed
  2005)

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Next Steps

• Goal is capturing the uncertainty/ biases/ etc.
  in these judgments
• Might need to quantify verbal expressions (e.g.,
  remote, likely, non-negligible..)
• Example if I say there is a negligible chance
  of anyone failing this class, what probability do
  you assume?
• What if I say non-negligible chance that someone
  will fail?

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Merging of Theories

• Science has known that objective and
  subjective factors existed for a long time
• Only more recently did we realize we could
  represent subjective as probabilities
• But inherently all of these subjective decisions
  can be ordered by decision tree
• Where we have a gamble or bet between what we
  know and what we think we know
• Clemen uses the basketball game gamble example
• We would keep adjusting payoffs until optimal

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Probability Wheel

• Mechanism for formalizing our thoughts on
  probabilities of comparative lotteries
• You select the area of the pie chart where until
  youre indifferent between the two lotteries
• Quick 2-person exercise. Then well discuss
  p-values.

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Heuristics and Biases

• Heuristics are rules of thumb
• Which do we use in life?
• Representativeness (fit category)
• Availability (seen it before, fits memory)
• Anchoring/Adjusting (common base point)
• Motivational Bias (perverse incentives)

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Continuous Distributions

• Similar to above, but we need to do it a few
  times.
• E.g., try to get 5, 50, 95 points on
  distribution

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Projects

• Groups of 3-4, others need permission
• Must have a real client (e.g., Brad/Don)
• Core model must be more than just cashflows
  (Decision Anal, MCDM, Monte Carlo,
  Cost-Effectiveness, etc.)
• Will have peer evaluations on effort/etc.
• Final product is a report of 15 pages
• Appendices, etc outside of 15 are ok
• Follow Writing Rubric (see syllabus)
• Next Wed Initial Groups, project outline with
  purpose, data sources, model, tasks (1 page)