Probability theory provides a framework for modeling and analyzing uncertainty — useful for statistics, stochastic processes, data science, and decision‑making.
Define sample space Ω, events as subsets of Ω. Probability function P assigning values in [0,1]. Axioms:
- P(Ω) = 1
- For disjoint A, B: P(A ∪ B) = P(A) + P(B)
- Probability of complement: P(Aᶜ) = 1 − P(A)
Definition: P(A|B) = P(A ∩ B)/P(B), provided P(B) > 0. Discuss independence: P(A∩B)=P(A)P(B), and consequences.
Random variable X: function from sample space to ℝ (or ℤ). Definition of expectation E[X], variance Var(X), for discrete distributions: E[X]=∑ x·P(X=x), Var(X)=E[(X−μ)²].
Define standard distributions. Show PMFs, expectation, variance. Useful for modeling random discrete phenomena: coin flips, counts, rare events.
Introduce idea that averages converge to expected value as sample size grows; for large enough samples, sum (or average) of many independent identically distributed random variables approaches a “bell‑shape” distribution. Mention approximation idea (without needing full rigorous measure theory here).
Probability theory describes uncertainty using:
- Ω – the sample space of all possible outcomes
- Events A, B ⊂ Ω – structured subsets of outcomes
- P – a probability measure assigning weights
Conditional probability, Bayes’ theorem, and random variables form the backbone of combinatorics, statistics, inference, and harmonic-modular systems.
🎲 Binomial Flip Visualizer
Simulate n coin flips (p = 0.5) and display histogram of heads over trials