Probabilities

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Probabilities

• Random Number Generators
• Actually pseudo-random
• Seed
• Same sequence from same seed
• Often time is used.
• Many examples on web.
• Custom random number generators exist.
• Can be used in algorithms.

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Actions Based on Probabilities

• Assign probabilities to each action.
• Probabilities must add to 1.
• Roulette wheel method
• Range of random number generator is size of
  roulette wheel.
• Give each action a section of the roulette wheel
  proportional to its probability.
• Generate a random number. (Run the wheel.)

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Example

• P(go straight) 50
• P(turn left) 30
• P(turn right) 20
• Use random numbers from 0 1.000
• Roulette wheel
• 0 Go straight lt .50
• .50 turn left lt .80
• .80 turn right lt 1.00

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What is Probability?

• Often not well defined.
• What does weather forecast of 75 rain mean?
• Calling your shots (dart board example).
• Interpretations
• Counting Interpretation
• Frequency Interpretation
• Subjective Interpretation

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Some Uses of Probability

• Diagnosis
• Prediction
• Explaining away
• water sprinkler example
• Randomized algorithms
• for CS in general
• for games and robotics in particular

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Expectation Value

• Expected value of a variable is a kind of average
  value of the variable.
• Sum of utilities times probability.
• Used in decision theory.
• Utility may be nonlinear.

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Assigning Subjective Probability

• Fair Bet
• Fair Price
• Dutch Book Fallacy
• Leads to probability rules

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Rules 1

• Values
• Real number between 0 and 1.
• Something happens
• P(something) 1
• Not rule
• P(not A) 1 P(A)

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Rules 2

• Or Rules P(A ? B)
• Exclusive events P(A) P(B)
• Not exclusive P(A) P(B) P(A ? B)
• no double counting
• And Rules P(A ? B)
• Independent events P(A) P(B)
• Conditional
• P(A B) P(B)
• P(B A) P(A)

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Bayes Theorem

• P(A) is prior probability of A
• P(A B) is posterior probability of A
• P(B) is prior probability of B acts as a
  normalizing constant
• Monte Hall Problem

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Bayesian Network

• Graph representing probabilistic causal relations
  between variables.
• Allows efficient Bayesian reasoning in
  complicated situations

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Simple Example

• Trapped ---? Locked
• 100 chests
• 37 trapped
• 29 of trapped were locked
• 63 not trapped
• 18 of not trapped locked
• Need to find P(trapped locked)

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