Lecture11-ReasoningAndDecisionMaking-1.pptx

All as are bs all bs are cs all as are cs some as are

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All A’s are B’s All B’s are C’s All A’s are C’s Some A’s are B’s Some B’s are C’s Some A’s are C’s 10
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Judge the validity of the following categorical syllogism: All the squares are striped Some of the striped objects have bold borders --------------------------------------------------------- Therefore, some of the squares have bold borders Another Type of Logical Structure: Categorical Syllogisms Is this syllogism true ? 11
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Most people mentally scan a few self-generated images like this one and conclude the syllogism is valid/true. Few people generate a large enough number of images, including this one, which might invalidate the syllogism. Johnson-Laird’s Mental Model Explanation for Errors With Categorical Syllogisms 12
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Deductive Versus Inductive Reasoning Deductive reasoning is concerned with conclusions that follow with certainty from their premises. Logical statements like conditional and categorical syllogisms rely on this type of reasoning. Inductive reasoning is concerned with conclusions that follow only probabilistically from their premises. 13
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Are humans better at inductive reasoning , perhaps because it is more “natural” to be less than completely certain about premises in our world? 14
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If a burglar is in the house, then the door will likely be ajar. Evidence (E): The door is ajar. What is the likelihood that my house has been burglarized (H)? Example of inductive reasoning: Bayes’ Theorem: An Objective Way to Calculate Probabilities for Inductive Reasoning 15
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Bayes’ Theorem: An Objective Way to Calculate Probabilities for Inductive Reasoning Three calculated probabilities: Prior Probability - Probability that the hypothesis is true before considering the evidence (H) or (~H) Conditional Probability - Probability that a particular piece of evidence is true if a particular hypothesis is or is not true (E|H) or (E|~H) Posterior Probability - Probability that the hypothesis is true given the evidence (this is what you ultimately want to figure out) ---> (H|E) 16
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Prior probability of a house in my neighborhood being burglarized (based on police reports): Conditional probability that the door of my house would be open if it had been burglarized: Conditional probability that the door of my house would be open if it had not been burglarized: Prob(H) = .002 (about 1 in 500) Prob(~H) = .998 or Prob(E|H) = .9 Prob(E|~H) = .01 An Application of Bayes’ Theorem 17
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Prob(H|E) = Prob(E|H) * Prob(H) Prob(E|H)*Prob(H) + Prob(E|~H)*Prob(~H) Posterior Probability Prior Probabilities Conditional Probabilities An Application of Bayes’ Theorem 18
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Prob(H|E) = Prob(E|H) * Prob(H) Prob(E|H)*Prob(H) + Prob(E|~H)*Prob(~H) Prob(H|E) = (.9) * (.002) (.9)*(.002) + (.01)*(.998) Prob(H|E) = .153 or 15.3% An Application of Bayes’ Theorem Plug in the prior and conditional probabilities in Bayes’ equation to get the posterior probability of house being burglarized given evidence of door being ajar: 19
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Prob(H|E) = Prob(E|H) * Prob(H) Prob(E|H)*Prob(H) + Prob(E|~H)*Prob(~H) Prob(H|E) = (.9) * (.002) (.9)*(.002) + (.01)*(.998) Does this seem low to you?
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