Introduction to Probability and Uncertainty

📋 Course Outline

1. Probability as a measure and function
2. Knightian and radical uncertainty
3. Random variables
4. Probability distributions and CDFs
5. Expected value, variance, and standard deviation
6. Uniform distribution

📖 1. Probability as a measure and function

🔑 Key Concepts & Definitions

• Sample space S: The sample space is the set of all possible outcomes of a random experiment.
• Power set 2^S: The power set is the collection of all subsets of S, including the empty set and S itself.
• Probability function: A probability function assigns a number between 0 and 1 to each event (subset of the sample space).
• Event: An event is a subset of the sample space that collects outcomes sharing a common property.

📝 Essential Points

• Probability is unsatisfactory when described only as chance or likelihood because it is vague and lacks a solid mathematical foundation.
• Probability is “like a measure” because it assigns numerical sizes to events similarly to how physical measurement assigns sizes to objects.
• The probability function must take events as inputs rather than a formula for one specific case.
• The probability function’s domain is 2^S and its output is a real number in the interval [0,1].
• Including both ∅ and S ensures probabilities can be assigned to the two extreme events.
• Probabilities larger than 1 would not have a sensible interpretation as degrees of certainty.

💡 Memory Hook