확률 — Probability & Statistics
Clean, honest explanations of probability concepts. No intimidation — just clarity.
Every complex probability concept starts with something simple. A coin flip. A die roll. A branching path. Learn the building blocks, and the rest follows naturally.
Enter values to see probabilities visualized in real-time.
Update your beliefs as new evidence arrives. P(A|B) tells you the probability of A, given that B has occurred. The foundation of modern inference.
P(A|B) = P(B|A) · P(A) / P(B)As you repeat an experiment more and more, the sample average converges to the expected value. The chaos settles into predictable patterns.
X̄n → μ as n → ∞No matter what distribution you start with, the average of many samples will be normally distributed. The bell curve emerges from chaos.
√n (X̄n - μ) / σ → N(0,1)The weighted average of all possible outcomes. Think of it as the balance point of a probability distribution — where the fulcrum would go.
E[X] = Σ xi · P(xi)How does knowing one event change the probability of another? Conditional probability narrows the sample space and recalculates.
P(A|B) = P(A ∩ B) / P(B)How spread out are the values? Variance measures the average squared distance from the mean. Standard deviation is its square root — in the same units as the data.
Var(X) = E[(X - μ)²]