by Albert Major

Critical and Scientific Thinking

Review of Reasoning Types

1. Deductive Reasoning
   • Provides certainty
   • Arguments are valid
   • Can be formalized
   • Most formal logic is deductive
2. Inductive Reasoning
   • Moving from specific cases to general conclusions
   • Example: Observing multiple black ravens to conclude all ravens are black
   • Not 100% certain but useful
   • Main focus of probability theory application
3. Abductive Reasoning (Inference to Best Explanation)
   • Used to find most likely explanation for observations
   • Example: Missing bike - more likely stolen than teleported by aliens
   • Common in everyday reasoning
   • Can be analyzed using probability

Probability Theory Applications

Base Rate Fallacy

The Base Rate Fallacy is when people focus on specific case information while ignoring the underlying statistical probability of something happening in the general population (like believing a witness who is 80% accurate without considering that there are far more green cabs than blue ones on the road).

• Example:

1. Cab Color Example:
   • Scenario:
     • 85% of cabs are green
     • 15% of cabs are blue
     • Witness identifies cab color with 80% accuracy
   • Key Finding:
     • Even with 80% accurate witness, probability of cab being blue is only 41%
     • Shows importance of considering base rates (total population)
   • Calculation:
     • Used Bayes' Theorem to show how base rates affect final probability
     • Demonstrates why intuitive judgments can be wrong
2. George's Study Case:
   • Given probabilities:
     • 4/5 chance George will study
     • 3/5 chance of passing if he studies
     • 1/10 chance of passing if he doesn't study
   • Demonstrates:
     • How to combine multiple probabilities
     • Importance of considering all scenarios

Bayes' Theorem

Bayes' Theorem is a mathematical formula that helps calculate the probability of a hypothesis being true by updating our initial beliefs with new evidence (like adjusting how likely we think all ravens are black each time we see another black raven).

1. Formula Components:

   P(A|B) = P(B|A) × P(A) / P(B)
   

   • P(A|B): Probability of A given B
   • P(B|A): Probability of B given A
   • P(A): Prior probability of A
   • P(B): Prior probability of B

2. Applications:

   • Updating beliefs based on new evidence
   • Evaluating competing hypotheses
   • Calculating conditional probabilities

• Example:

Scientific Reasoning and Probability

Ravens Example