확률

The Art of Probability

Where mathematics meets elegance

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Fundamental Concepts

P(A) 0.50

Classical Probability

The ratio of favorable outcomes to total outcomes. The foundation upon which all probability theory is built.

P(μ±σ) 0.68

Normal Distribution

The bell curve that governs natural phenomena. 68% of values fall within one standard deviation of the mean.

P(B|A) 0.37

Bayes' Theorem

Updating beliefs with evidence. The probability of a hypothesis given observed data transforms our understanding.

P(LLN) 0.95

Law of Large Numbers

As trials increase, the sample mean converges to the expected value. Certainty emerges from randomness.

Distribution Visualizer

Mean (μ) 0.00
Std Dev (σ) 1.00
Variance (σ²) 1.00

Probability Experiment

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Essential Formulas

P

Probability

P(A) = n(A) / n(S)

Favorable outcomes divided by total sample space

n! r!(n-r)!

Combinations

C(n,r) = n! / r!(n-r)!

Number of ways to choose r items from n without order

Expected Value

E(X) = Σ xᵢ · P(xᵢ)

The weighted average of all possible outcomes

Bayes' Theorem

P(A|B) = P(B|A)·P(A) / P(B)

Posterior probability from prior and likelihood