Probability Formulas — Complete Reference Sheet A complete reference of probability formulas — basic rules, counting, conditional probability, Bayes' theorem, expected value, variance, and the major discrete and continuous distributions. Each formula links to its full page where available.
Basic Probability Probability of an Event
P ( A ) = favorable outcomes total outcomes P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}} P ( A ) = total outcomes favorable outcomes Complement Rule
P ( A c ) = 1 − P ( A ) P(A^c) = 1 - P(A) P ( A c ) = 1 − P ( A ) Addition Rule (Any Events)
P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) P(A \cup B) = P(A) + P(B) - P(A \cap B) P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) Addition Rule (Mutually Exclusive)
P ( A ∪ B ) = P ( A ) + P ( B ) P(A \cup B) = P(A) + P(B) P ( A ∪ B ) = P ( A ) + P ( B ) Multiplication Rule (Independent)
P ( A ∩ B ) = P ( A ) ⋅ P ( B ) P(A \cap B) = P(A) \cdot P(B) P ( A ∩ B ) = P ( A ) ⋅ P ( B ) Multiplication Rule (General)
P ( A ∩ B ) = P ( A ) ⋅ P ( B ∣ A ) P(A \cap B) = P(A) \cdot P(B \mid A) P ( A ∩ B ) = P ( A ) ⋅ P ( B ∣ A ) Conditional Probability & Bayes' Theorem Conditional Probability
P ( A ∣ B ) = P ( A ∩ B ) P ( B ) P(A \mid B) = \frac{P(A \cap B)}{P(B)} P ( A ∣ B ) = P ( B ) P ( A ∩ B ) Bayes' Theorem
P ( A ∣ B ) = P ( B ∣ A ) P ( A ) P ( B ) P(A \mid B) = \frac{P(B \mid A)\,P(A)}{P(B)} P ( A ∣ B ) = P ( B ) P ( B ∣ A ) P ( A ) Law of Total Probability
P ( B ) = ∑ i P ( B ∣ A i ) P ( A i ) P(B) = \sum_i P(B \mid A_i)\,P(A_i) P ( B ) = i ∑ P ( B ∣ A i ) P ( A i ) Independence Test
A , B independent ⟺ P ( A ∩ B ) = P ( A ) P ( B ) A,B \text{ independent} \iff P(A \cap B) = P(A) P(B) A , B independent ⟺ P ( A ∩ B ) = P ( A ) P ( B ) Counting (Permutations & Combinations) Factorial
n ! = n ( n − 1 ) ( n − 2 ) ⋯ 1 n! = n(n-1)(n-2)\cdots 1 n ! = n ( n − 1 ) ( n − 2 ) ⋯ 1 Permutations
n P r = n ! ( n − r ) ! {}_n P_r = \frac{n!}{(n-r)!} n P r = ( n − r )! n ! Combinations
n C r = ( n r ) = n ! r ! ( n − r ) ! {}_n C_r = \binom{n}{r} = \frac{n!}{r!(n-r)!} n C r = ( r n ) = r ! ( n − r )! n ! Fundamental Counting Principle
N = n 1 ⋅ n 2 ⋯ n k N = n_1 \cdot n_2 \cdots n_k N = n 1 ⋅ n 2 ⋯ n k Permutations with Repetition
n ! n 1 ! n 2 ! ⋯ n k ! \frac{n!}{n_1!\,n_2!\,\cdots\,n_k!} n 1 ! n 2 ! ⋯ n k ! n ! Expected Value, Variance & SD Expected Value (Discrete)
E [ X ] = ∑ i x i P ( x i ) E[X] = \sum_i x_i\,P(x_i) E [ X ] = i ∑ x i P ( x i ) Expected Value (Continuous)
E [ X ] = ∫ − ∞ ∞ x f ( x ) d x E[X] = \int_{-\infty}^{\infty} x\,f(x)\,dx E [ X ] = ∫ − ∞ ∞ x f ( x ) d x Variance
Var ( X ) = E [ ( X − μ ) 2 ] = E [ X 2 ] − μ 2 \operatorname{Var}(X) = E[(X - \mu)^2] = E[X^2] - \mu^2 Var ( X ) = E [( X − μ ) 2 ] = E [ X 2 ] − μ 2 Standard Deviation
σ = Var ( X ) \sigma = \sqrt{\operatorname{Var}(X)} σ = Var ( X ) Linear Transform
E [ a X + b ] = a E [ X ] + b , Var ( a X + b ) = a 2 Var ( X ) E[aX + b] = a E[X] + b,\quad \operatorname{Var}(aX + b) = a^2 \operatorname{Var}(X) E [ a X + b ] = a E [ X ] + b , Var ( a X + b ) = a 2 Var ( X ) Discrete Distributions Bernoulli
P ( X = 1 ) = p , P ( X = 0 ) = 1 − p P(X = 1) = p,\ P(X = 0) = 1-p P ( X = 1 ) = p , P ( X = 0 ) = 1 − p Binomial
P ( X = k ) = ( n k ) p k ( 1 − p ) n − k P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} P ( X = k ) = ( k n ) p k ( 1 − p ) n − k Binomial Mean / Var
μ = n p , σ 2 = n p ( 1 − p ) \mu = np,\quad \sigma^2 = np(1-p) μ = n p , σ 2 = n p ( 1 − p ) Geometric (first success)
P ( X = k ) = ( 1 − p ) k − 1 p P(X = k) = (1-p)^{k-1} p P ( X = k ) = ( 1 − p ) k − 1 p Poisson
P ( X = k ) = λ k e − λ k ! P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} P ( X = k ) = k ! λ k e − λ Hypergeometric
P ( X = k ) = ( K k ) ( N − K n − k ) ( N n ) P(X = k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}} P ( X = k ) = ( n N ) ( k K ) ( n − k N − K ) Continuous Distributions Uniform on [a, b]
f ( x ) = 1 b − a , a ≤ x ≤ b f(x) = \frac{1}{b-a},\ a \le x \le b f ( x ) = b − a 1 , a ≤ x ≤ b Normal (Gaussian)
f ( x ) = 1 σ 2 π e − ( x − μ ) 2 / ( 2 σ 2 ) f(x) = \frac{1}{\sigma\sqrt{2\pi}}\,e^{-(x-\mu)^2 / (2\sigma^2)} f ( x ) = σ 2 π 1 e − ( x − μ ) 2 / ( 2 σ 2 ) Standard Normal Z-score
Z = X − μ σ Z = \frac{X - \mu}{\sigma} Z = σ X − μ Exponential
f ( x ) = λ e − λ x , x ≥ 0 f(x) = \lambda e^{-\lambda x},\ x \ge 0 f ( x ) = λ e − λ x , x ≥ 0 Exponential Mean / Var
μ = 1 λ , σ 2 = 1 λ 2 \mu = \tfrac{1}{\lambda},\quad \sigma^2 = \tfrac{1}{\lambda^2} μ = λ 1 , σ 2 = λ 2 1 