Discrete Probability Summary

Discrete Math · Axiom Academy

Let's review how probability provides a mathematical framework for quantifying uncertainty and making decisions under incomplete information. Sample Space: The set of all possible outcomes of an experiment, denoted S or Ω Event: Any subset of the sample space. Probability assigns a value between 0 and 1 to each event Probability Function: Must satisfy: P(S) = 1 , P(A) ≥ 0 for all events, and P(A ∪ B) = P(A) + P(B) for disjoint events Law of Large Numbers: As trials increase, experimental probability converges to theoretical probability Equally Likely Outcomes: When all outcomes have equal probability, P(A) = |A| / |S| Complement Rule: P(A c ) = 1 - P(A) , useful when counting what doesn't happen is easier Addition Rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) for any events A and B Multiplication Principle: Combine permutations and combinations with probability for complex experiments Definition: Probability of A given B occurred: P(A|B) = P(A ∩ B) / P(B) when P(B) > 0 Updated Beliefs: Conditional probability formalizes how new information changes our assessment of likelihood Independence: Events A and B are independent if P(A ∩ B) = P(A) · P(B) , equivalently P(A|B) = P(A) Multiplication Rule: P(A ∩ B) = P(A) · P(B|A) , extends to chains of events in tree diagrams Definition: A function X: S → ℝ that assigns numerical values to outcomes PMF: Probability mass function p(x) = P(X = x) specifies the distribution

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