knowledge/docs/probability.txt

PROBABILITY FORMULAS AND CONCEPTS

Basic Probability:
- P(A) = favorable outcomes / total outcomes
- 0 ≤ P(A) ≤ 1
- P(A') = 1 - P(A)

Addition Rules:
- P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
- For mutually exclusive: P(A ∪ B) = P(A) + P(B)

Multiplication Rules:
- P(A ∩ B) = P(A) × P(B|A)
- For independent: P(A ∩ B) = P(A) × P(B)

Conditional Probability:
- P(A|B) = P(A ∩ B) / P(B)
- Bayes' Theorem: P(A|B) = P(B|A) × P(A) / P(B)

Combinatorics:
- Permutation: nPr = n! / (n-r)!
- Combination: nCr = n! / (r!(n-r)!)
- Circular permutation: (n-1)!

Distributions:
- Binomial: P(X=k) = nCk × p^k × (1-p)^(n-k)
- Expected value: E(X) = Σ x × P(x)
- Variance: Var(X) = E(X²) - [E(X)]²

Common Mistakes:
- Confusing P(A|B) with P(B|A)
- Forgetting to account for replacement
- Not checking if events are independent