Every event has a probability between 0 and 1. The sum of all possible outcomes equals certainty. From this simple foundation, an entire mathematical universe unfolds.
P(A) = n(A) / n(S)The probability of A given that B has occurred. Conditional probability is where intuition meets mathematics, and where Bayes showed us how to update our beliefs.
P(A|B) = P(A∩B) / P(B)Two events are independent when knowing one tells you nothing about the other. The coin does not remember its previous flip. Each moment begins fresh.
P(A∩B) = P(A) × P(B)The engine of rational inference. Bayes' theorem tells us how to update our beliefs in light of new evidence. It is the mathematical foundation of learning from experience.
P(A|B) = P(B|A)P(A) / P(B)As the number of trials increases, the experimental probability converges to the theoretical probability. Patience reveals truth. The universe is fair in the limit.
lim(n→∞) X̄ₙ = μProbability is not about predicting the future. It is about understanding uncertainty with honesty and precision. The beauty of probability is the beauty of humility before complexity.
∑ P(xᵢ) = 1