probability

name: probability description: Random events and likelihood license: MIT compatibility: opencode metadata: audience: mathematicians category: mathematics

What I do

• Calculate probabilities and conditional probabilities
• Apply Bayes' theorem for inference
• Work with probability distributions (discrete and continuous)
• Calculate expected values and variances
• Apply the central limit theorem
• Use moment generating functions

When to use me

When analyzing random phenomena, making inferences from data, or modeling uncertainty.

Key Concepts

• Probability Axioms: P(A) ≥ 0, P(S) = 1, P(∪A_i) = ΣP(A_i) for disjoint events
• Conditional Probability: P(A|B) = P(A∩B)/P(B)
• Bayes' Theorem: P(A|B) = P(B|A)P(A)/P(B)
• Expected Value: E[X] = Σ x·P(X=x) or ∫x·f(x)dx
• Variance: Var(X) = E[X²] - (E[X])²
• Central Limit Theorem: Sum of i.i.d. variables approaches normal distribution