📐 Core Equations with Excel Examples
1) Set Laws & Probability Basics
Complement: P(A′) = 1 − P(A)
Union: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Intersection: P(A ∩ B) = P(A|B) · P(B)
All probabilities satisfy 0 ≤ P(event) ≤ 1, and total probability = 1.
Excel: =1 - P(A)
Excel: =P(A) + P(B) - P(A_and_B)
Excel: =P_A_given_B * P_B
2) Counting Rules
Multiplication Rule: N = n1 × n2 × ... × nk
Permutations: P(n,r) = n! / (n − r)!
Combinations: C(n,r) = n! / [r!(n − r)!]
Order matters? Use permutations. Order doesn’t matter? Use combinations.
Excel: =PERMUT(n,r)
Excel: =COMBIN(n,r)
Excel: =FACT(n)
3) Addition & Multiplication
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
P(A ∩ B) = P(A|B) · P(B)
If independent: P(A ∩ B) = P(A) · P(B)
Excel: =P_A_given_B * P_B
Excel: =P_A * P_B
4) Conditional Probability
P(A|B) = P(A ∩ B) / P(B)
Excel: =P_A_and_B / P_B
5) Total Probability Rule
P(B) = Σ P(B|Ei) · P(Ei)
Excel (2 scenarios): =P_B_given_E1*P_E1 + P_B_given_E2*P_E2
6) Bayes’ Theorem (2-event)
P(E1|B) =
[ P(B|E1) · P(E1) ] ÷
[ P(B|E1) · P(E1) + P(B|E2) · P(E2) ]
Posterior = (Likelihood × Prior) ÷ (Total Probability of Evidence)
Numerator =P_B_given_E1 * P_E1
Denominator =P_B_given_E1*P_E1 + P_B_given_E2*P_E2
Bayes =Numerator / Denominator
7) Independence
Independent if P(A ∩ B) = P(A) · P(B)
Equivalent: P(A|B) = P(A), and P(B|A) = P(B)
Excel check:
=IF(ABS(P_A_and_B - (P_A*P_B)) < 0.0001,"Independent","Not Independent")