Probability Basics

Math for CS · Axiom Academy

Axioms, rules, and fundamental operations A probability is a number between 0 and 1 assigned to an event that quantifies how likely the event is to occur. We write P(A) for the probability of event A. The mathematical foundation of probability rests on three axioms formulated by Andrey Kolmogorov: Probabilities are never negative. The probability of the entire sample space is 1. Something must happen. The probability of a union of disjoint events is the sum of their individual probabilities. The probability of NOT A is one minus the probability of A. The empty set (impossible event) has probability zero. If A is a subset of B, A cannot be more likely than B. We subtract P(A ∩ B) to avoid double-counting outcomes in both A and B. Rule 5: Union of Disjoint Events When events don't overlap, the union rule simplifies. For finite sample spaces with equally likely outcomes, we compute probabilities by counting: Many problems require counting arrangements: For experiments with two independent choices, probability distributions can be displayed in a table: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) If A and B are disjoint: P(A ∪ B) = P(A) + P(B) P(A) = |A| / |Ω| for equally likely outcomes Axiom Academy | Probability for Computer Science

This is the written version of the interactive lesson above. See the full Math for CS course.