Classical: P(A) = n(A)/n(S), P(A') = 1-P(A) Addition: P(A union B) = P(A)+P(B)-P(A intersect B) Conditional: P(A|B) = P(A intersect B)/P(B) Multiplication: P(A intersect B) = P(A)*P(B|A) Independence: P(A intersect B) = P(A)*P(B) Total Probability: P(A) = sum P(A|B_i)*P(B_i) Bayes': P(B_i|A) = P(A|B_i)*P(B_i)/sum P(A|B_j)*P(B_j)

Binomial Distribution X ~ B(n,p):

• PMF: P(X=r) = C(n,r)p^rq^(n-r)
• Mean: E(X) = np
• Variance: Var(X) = npq
• Mode: floor((n+1)p) or two values if (n+1)p is integer
• Sum: sum P(X=r) = 1

Inclusion-Exclusion (3 events): P(A union B union C) = P(A)+P(B)+P(C)-P(AB)-P(BC)-P(AC)+P(ABC)

Counting:

• Permutations: nPr = n!/(n-r)!
• Combinations: nCr = n!/(r!(n-r)!)
• With replacement: n^r

Expected Value: E(aX+b) = aE(X)+b Variance: Var(aX+b) = a^2*Var(X)

Geometric: P(X=k) = q^(k-1)*p (first success on kth trial)