Probability

P(A|B) = probability of A given B has occurred = P(A∩B)/P(B), where P(B)>0. Reduces sample space to events consistent with B. Example: P(even number | >2) on a die: favorable = {4,6}, total from >2 is {3,4,5,6}, so P = 2/4 = 1/2.

P(A∩B) = P(A) × P(B|A) = P(B) × P(A|B). For independent events: P(A∩B) = P(A) × P(B). Events A and B are independent if: P(A|B) = P(A) or P(B|A) = P(B). Mutually exclusive vs independent: ME means P(A∩B)=0, independent means P(A∩B)=P(A)P(B).

P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ) / Σⱼ P(Aⱼ)P(B|Aⱼ). Used for: medical diagnosis (P(disease|+test)), quality control, spam filters. Partition theorem: P(B) = ΣP(B|Aᵢ)P(Aᵢ). Example: 3 boxes with different compositions — which box was a ball drawn from?

Random variable X assigns numerical value to each outcome. Mean (expectation) E(X)=Σx⋅P(X=x). Variance V(X)=E(X²)-[E(X)]². Binomial distribution: n trials, each with success probability p. P(X=r) = ⁿCᵣ × pʳ × (1-p)^(n-r). Mean = np. Variance = npq where q=1-p. Used in: coin tosses, quality testing, genetics.