← 🎲 Probability

🎲 Sample Space Ω

|A| = · P(A) =
|B| = · P(B) =
|A∩B| = · P(A∩B) =
|A∪B| = · P(A∪B) =
Rolls: 0
Empirical freq. in view:
Axiom 1: P(E) ≥ 0 for every event.
Axiom 2: P(Ω) = 1, P(∅) = 0.
Axiom 3: for disjoint A, B — P(A∪B) = P(A) + P(B).
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🎲 Sample Space, Events, and the Kolmogorov Axioms

The 6×6 grid of tiles is the full sample space Ω for rolling two dice — 36 equally likely ordered outcomes. Choosing events raises and colours the subsets of Ω that belong to them, turning abstract set theory into something you can count directly.

🔬 What It Demonstrates

Events are subsets of a sample space; probability is the fraction of Ω they cover. Union and intersection combine subsets, and the axioms — non-negativity, P(Ω)=1, and additivity over disjoint events — fall directly out of counting tiles.

🎮 How to Use

Pick Event A and Event B, then choose whether to view A alone, B alone, their union, or their intersection. Roll the dice (or enable auto-roll) to sample outcomes and watch the empirical frequency converge on the theoretical probability shown for the region.

💡 Did You Know?

Kolmogorov's three axioms — proposed in 1933 — are the entire foundation of modern probability theory. Every theorem, including Bayes' rule and the law of large numbers, is provable from just those three statements about sets.