Reasoning About Probabilities Overview

Reasoning About Probabilities

• Prior Probability is the probability of a statement being true before considering evidence.
• Posterior Probability is the probability of a statement being true after considering evidence.
• To calculate posterior probability, we need prior probability (base rate), evidence, and reliability of the evidence.
• Bayes’ Theorem calculates posterior probability.
• Base-rate neglect occurs when people ignore base rates when making probability judgments.
• Hammerton (1973) studied base-rate neglect in a rare cancer diagnostic test scenario.
• People tend to overestimate the probability of having cancer after a positive test result, neglecting the low base rate of the disease.
• Tversky and Kahneman (1974) found that people overestimate the frequency of words starting with the letter "k" compared to those having "k" in the third position.

Problem Solving, Reasoning, & Decision Making

• Problem solving involves problem space and search, operator acquisition and selection, and problem representation.
• Reasoning and decision-making involve reasoning about conditionals, reasoning about probabilities, and decision making with subjective utility.

Decision Making

• The Wason Selection Task demonstrates how people struggle to make logically correct choices in probabilistic scenarios.
• People tend to interpret conditional statements based on probabilistic models rather than strict logical models.
• This might explain why the logically correct choice in the Wason Selection Task is so difficult.

Problem Representation

• Problem representation significantly affects problem-solving success.
• Successful problem-solving requires representing the problem in a way that allows relevant operators to be applied.
• The Mutilated Checkerboard Problem illustrates how representing a problem differently can lead to a solution.
• Incubation effects occur when solutions become easier to find after a period of time spent away from the problem.
• Silveira's (1971) Cheap-Necklace Problem demonstrates that incubation effects can lead to increased problem-solving success.

Reasoning and Decision Making

• Humans often make illogical conclusions in logical problems. However, computer systems based on formal logic make errors humans wouldn't.
• Four areas where human irrationality is often found: reasoning about conditionals, reasoning about quantifiers, reasoning about probabilities, and decision making.

Reasoning About Conditionals

• A conditional statement expresses an assertion that if an antecedent (A) is true, then a consequent (B) must be true.
• Wason's Selection Task is a classic example of how people struggle to apply logical rules to conditional statements.
• Participants are given cards with letters and numbers and asked to choose the cards that need to be turned over to confirm the rule: "If a card has a vowel on one side, then it has an even number on the other side."