Free SOA Exam P (Probability) Formula Sheet (2026)

Every Exam P formula you need on the test, grouped by topic, rendered with full math notation. 67 formulas across 8 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

67 Formulas
8 Topics
2026 Syllabus
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All Exam P Formulas

Probability Fundamentals 5 items
Set complement
P(Ac)=1−P(A)P(A^c) = 1 - P(A)
Addition rule (two events)
P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)
Addition rule (three events)
P(A∪B∪C)=P(A)+P(B)+P(C)−P(A∩B)−P(A∩C)−P(B∩C)+P(A∩B∩C)P(A \cup B \cup C) = P(A)+P(B)+P(C) - P(A\cap B) - P(A\cap C) - P(B\cap C) + P(A\cap B\cap C)
Permutations (ordered)
nPr=n!(n−r)!_nP_r = \dfrac{n!}{(n-r)!}
nn objects taken rr at a time, order matters
Combinations (unordered)
(nr)=n!r! (n−r)!\binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}
nn objects taken rr at a time, order does not matter
Conditional Probability & Bayes 5 items
Conditional probability
P(A∣B)=P(A∩B)P(B),P(B)>0P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}, \quad P(B)>0
Multiplication rule
P(A∩B)=P(A∣B) P(B)=P(B∣A) P(A)P(A \cap B) = P(A \mid B)\,P(B) = P(B \mid A)\,P(A)
Law of total probability
P(A)=∑iP(A∣Bi) P(Bi)P(A) = \sum_{i} P(A \mid B_i)\,P(B_i)
where {Bi}\{B_i\} is a partition of the sample space
Bayes' theorem
P(Bi∣A)=P(A∣Bi) P(Bi)∑jP(A∣Bj) P(Bj)P(B_i \mid A) = \dfrac{P(A \mid B_i)\,P(B_i)}{\sum_j P(A \mid B_j)\,P(B_j)}
Independence of two events
AA and BB are independent iff P(A∩B)=P(A) P(B)P(A \cap B) = P(A)\,P(B)
(equivalently: P(A∣B)=P(A)P(A\mid B)=P(A))
Discrete Distributions 6 items
Bernoulli distribution — PMF, mean, variance
P(X=1)=p,  P(X=0)=1−pP(X=1)=p,\; P(X=0)=1-p
E[X]=p,Var(X)=p(1−p)E[X]=p,\quad \text{Var}(X)=p(1-p)
Binomial distribution — PMF, mean, variance
P(X=k)=(nk)pk(1−p)n−kP(X=k)=\binom{n}{k}p^k(1-p)^{n-k}
E[X]=np,Var(X)=np(1−p)E[X]=np,\quad \text{Var}(X)=np(1-p)
k=0,1,…,nk=0,1,\ldots,n
Poisson distribution — PMF, mean, variance
P(X=k)=e−λλkk!P(X=k)=\dfrac{e^{-\lambda}\lambda^k}{k!}
E[X]=λ,Var(X)=λE[X]=\lambda,\quad \text{Var}(X)=\lambda
k=0,1,2,…k=0,1,2,\ldots
Geometric distribution — PMF, mean, variance
P(X=k)=(1−p)k−1pP(X=k)=(1-p)^{k-1}p (number of trials to first success)
E[X]=1p,Var(X)=1−pp2E[X]=\dfrac{1}{p},\quad \text{Var}(X)=\dfrac{1-p}{p^2}
k=1,2,…k=1,2,\ldots
Negative Binomial distribution — PMF, mean, variance
P(X=k)=(k−1r−1)pr(1−p)k−rP(X=k)=\binom{k-1}{r-1}p^r(1-p)^{k-r} (trials for rrth success)
E[X]=rp,Var(X)=r(1−p)p2E[X]=\dfrac{r}{p},\quad \text{Var}(X)=\dfrac{r(1-p)}{p^2}
Hypergeometric distribution — PMF, mean, variance
P(X=k)=(Kk)(N−Kn−k)(Nn)P(X=k)=\dfrac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}
E[X]=nKN,Var(X)=nK(N−K)(N−n)N2(N−1)E[X]=\dfrac{nK}{N},\quad \text{Var}(X)=\dfrac{nK(N-K)(N-n)}{N^2(N-1)}
Continuous Distributions 8 items
Uniform distribution — PDF, mean, variance
f(x)=1b−a,a<x<bf(x)=\dfrac{1}{b-a},\quad a<x<b
E[X]=a+b2,Var(X)=(b−a)212E[X]=\dfrac{a+b}{2},\quad \text{Var}(X)=\dfrac{(b-a)^2}{12}
Exponential distribution — PDF, CDF, mean, variance
f(x)=λe−λx,F(x)=1−e−λx,x>0f(x)=\lambda e^{-\lambda x},\quad F(x)=1-e^{-\lambda x},\quad x>0
E[X]=1λ,Var(X)=1λ2E[X]=\dfrac{1}{\lambda},\quad \text{Var}(X)=\dfrac{1}{\lambda^2}
Memoryless property (Exponential / Geometric)
P(X>s+t∣X>s)=P(X>t)P(X > s+t \mid X > s) = P(X > t)
Only the Exponential (continuous) and Geometric (discrete) satisfy this.
Normal distribution — PDF
f(x)=1σ2πexp⁡ ⁣(−(x−μ)22σ2)f(x)=\dfrac{1}{\sigma\sqrt{2\pi}}\exp\!\left(-\dfrac{(x-\mu)^2}{2\sigma^2}\right)
E[X]=μ,Var(X)=σ2E[X]=\mu,\quad \text{Var}(X)=\sigma^2
Gamma distribution — PDF, mean, variance
f(x)=xα−1e−x/θΓ(α) θα,x>0f(x)=\dfrac{x^{\alpha-1}e^{-x/\theta}}{\Gamma(\alpha)\,\theta^\alpha},\quad x>0
E[X]=αθ,Var(X)=αθ2E[X]=\alpha\theta,\quad \text{Var}(X)=\alpha\theta^2
α\alpha=shape, θ\theta=scale
Weibull distribution — PDF, mean
f(x)=τθ(xθ)τ−1e−(x/θ)τ,x>0f(x)=\dfrac{\tau}{\theta}\left(\dfrac{x}{\theta}\right)^{\tau-1}e^{-(x/\theta)^\tau},\quad x>0
E[X]=θ Γ ⁣(1+1τ)E[X]=\theta\,\Gamma\!\left(1+\tfrac{1}{\tau}\right)
Lognormal distribution — PDF, mean, variance
f(x)=1xσ2πexp⁡ ⁣(−(ln⁡x−μ)22σ2),x>0f(x)=\dfrac{1}{x\sigma\sqrt{2\pi}}\exp\!\left(-\dfrac{(\ln x-\mu)^2}{2\sigma^2}\right),\quad x>0
E[X]=eμ+σ2/2,Var(X)=e2μ+σ2(eσ2−1)E[X]=e^{\mu+\sigma^2/2},\quad \text{Var}(X)=e^{2\mu+\sigma^2}(e^{\sigma^2}-1)
Beta distribution — PDF, mean
f(x)=xα−1(1−x)β−1B(α,β),0<x<1f(x)=\dfrac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)},\quad 0<x<1
E[X]=αα+β,Var(X)=αβ(α+β)2(α+β+1)E[X]=\dfrac{\alpha}{\alpha+\beta},\quad \text{Var}(X)=\dfrac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}
Joint & Marginal Distributions 3 items
Joint PDF — marginal densities
fX(x)=∫−∞∞fX,Y(x,y) dyf_X(x)=\int_{-\infty}^{\infty}f_{X,Y}(x,y)\,dy
fY(y)=∫−∞∞fX,Y(x,y) dxf_Y(y)=\int_{-\infty}^{\infty}f_{X,Y}(x,y)\,dx
Conditional PDF
fY∣X(y∣x)=fX,Y(x,y)fX(x)f_{Y\mid X}(y\mid x)=\dfrac{f_{X,Y}(x,y)}{f_X(x)}
Independence of continuous RVs
X,YX,Y independent iff fX,Y(x,y)=fX(x) fY(y)f_{X,Y}(x,y)=f_X(x)\,f_Y(y) for all x,yx,y
Expectation & Variance 7 items
MGF definition and moment extraction
MX(t)=E[etX]M_X(t)=E[e^{tX}]
E[Xn]=MX(n)(0)E[X^n]=M_X^{(n)}(0) (nnth derivative at t=0t=0)
Variance shortcut
Var(X)=E[X2]−(E[X])2\text{Var}(X)=E[X^2]-(E[X])^2
Covariance
Cov(X,Y)=E[XY]−E[X] E[Y]\text{Cov}(X,Y)=E[XY]-E[X]\,E[Y]
If X,YX,Y independent: Cov(X,Y)=0\text{Cov}(X,Y)=0
Correlation coefficient
ρXY=Cov(X,Y)σX σY,−1≤ρ≤1\rho_{XY}=\dfrac{\text{Cov}(X,Y)}{\sigma_X\,\sigma_Y},\quad -1\le\rho\le1
Variance of a linear combination
Var(aX+bY)=a2Var(X)+b2Var(Y)+2ab Cov(X,Y)\text{Var}(aX+bY)=a^2\text{Var}(X)+b^2\text{Var}(Y)+2ab\,\text{Cov}(X,Y)
Law of total expectation
E[X]=E[E[X∣Y]]E[X]=E[E[X\mid Y]]
Law of total variance
Var(X)=E[Var(X∣Y)]+Var(E[X∣Y])\text{Var}(X)=E[\text{Var}(X\mid Y)]+\text{Var}(E[X\mid Y])
Insurance Applications 5 items
Ordinary deductible — payment per loss
YL=max⁡(X−d, 0)Y^L = \max(X-d,\,0)
E[YL]=E[X]−E[X∧d]E[Y^L]=E[X]-E[X\wedge d]
where dd = deductible, XX = ground-up loss
Limited expected value (LEV)
E[X∧u]=∫0uS(x) dx=∫0u[1−F(x)] dxE[X\wedge u]=\int_0^u S(x)\,dx = \int_0^u [1-F(x)]\,dx
for X≥0X\ge 0
Payment per payment (excess loss)
e(d)=E[X−d∣X>d]=E[X]−E[X∧d]1−F(d)e(d)=E[X-d\mid X>d]=\dfrac{E[X]-E[X\wedge d]}{1-F(d)}
dd = deductible
Policy limit — payment per loss
YL=min⁡(X, u)=X∧uY^L=\min(X,\,u)=X\wedge u
E[YL]=E[X∧u]E[Y^L]=E[X\wedge u]
uu = policy limit
Stop-loss (aggregate) premium
E[(S−d)+]=E[S]−E[S∧d]E[(S-d)_+]=E[S]-E[S\wedge d]
SS = aggregate loss, dd = retention (aggregate deductible)
Topic 0 28 items
De Morgan's Laws
(A∪B)c=Ac∩Bc(A \cup B)^c = A^c \cap B^c and (A∩B)c=Ac∪Bc(A \cap B)^c = A^c \cup B^c; A, B = events, c^c = complement, ∪\cup = union, ∩\cap = intersection
Stars and bars distribution count
(n+k−1k−1)\binom{n+k-1}{k-1}, n = identical objects distributed, k = distinct bins (non-negative integer solutions to x1+⋯+xk=nx_1+\cdots+x_k=n); use (n−1k−1)\binom{n-1}{k-1} if each bin needs at least one
Addition rule for mutually exclusive events
P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B) when P(A∩B)=0P(A \cap B) = 0; A, B = mutually exclusive (disjoint) events that cannot both occur
Feasibility bounds for an intersection probability
max⁡(0,P(A)+P(B)−1)≤P(A∩B)≤min⁡(P(A),P(B))\max(0, P(A)+P(B)-1) \leq P(A \cap B) \leq \min(P(A), P(B)), lower bound = Bonferroni, upper bound from subset containment
Chain rule for the probability of an intersection of events
P(A1∩⋯∩An)=P(A1)P(A2∣A1)P(A3∣A1∩A2)⋯P(An∣A1∩⋯∩An−1)P(A_1 \cap \cdots \cap A_n) = P(A_1)P(A_2 \mid A_1)P(A_3 \mid A_1 \cap A_2)\cdots P(A_n \mid A_1 \cap \cdots \cap A_{n-1}), each factor conditions on all prior events
Odds form of Bayes' theorem
P(B∣A)P(Bc∣A)=P(A∣B)P(A∣Bc)⋅P(B)P(Bc)\frac{P(B \mid A)}{P(B^c \mid A)} = \frac{P(A \mid B)}{P(A \mid B^c)} \cdot \frac{P(B)}{P(B^c)}, posterior odds = likelihood ratio × prior odds; A = evidence, B = hypothesis, B^c = complement
Distinguishable permutations of a multiset
N=n!n1! n2!⋯nk! N = \dfrac{n!}{n_1!\,n_2!\cdots n_k!} , n = total items, n_i = count of identical items of type i, k = number of distinct types
Survival function in terms of the CDF
SX(x)=P(X>x)=1−FX(x)S_X(x) = P(X > x) = 1 - F_X(x), SXS_X = survival function, FX(x)=P(X≤x)F_X(x)=P(X\le x) = CDF, x = threshold value
Conditional CDF of a left-truncated distribution
FX∣X>a(x)=FX(x)−FX(a)1−FX(a), x>aF_{X|X>a}(x) = \frac{F_X(x) - F_X(a)}{1 - F_X(a)}, \ x > a, F = CDF, a = truncation point, x = value
Law of the Unconscious Statistician (LOTUS)
E[g(X)]=∫−∞∞g(x)fX(x) dxE[g(X)] = \int_{-\infty}^{\infty} g(x) f_X(x)\, dx, g = function of X, f_X = pdf of X (use sum with p_X(x) if discrete)
Chebyshev's inequality
P(∣X−μ∣≥kσ)≤1k2P(|X - \mu| \geq k\sigma) \leq \frac{1}{k^2}, μ = mean, σ = standard deviation, k = number of SDs; equivalently at least 1−1/k21 - 1/k^2 lies within k SDs
Expected payment per loss with deductible and benefit limit
E[YL]=E[X∧(d+u)]−E[X∧d]E[Y^L] = E[X \wedge (d+u)] - E[X \wedge d], d = deductible, u = benefit limit, X = ground-up loss, X∧aX \wedge a = loss capped at a
Second moment of the excess loss for an exponential
E[((X−d)+)2]=S(d)⋅2θ2=e−d/θ⋅2θ2E[((X-d)_+)^2] = S(d)\cdot 2\theta^2 = e^{-d/\theta}\cdot 2\theta^2, d = deductible, θ = exponential mean, S(d) = survival function at d
Pareto distribution pdf, mean, and variance
f(x)=αθα(x+θ)α+1f(x) = \frac{\alpha\theta^\alpha}{(x+\theta)^{\alpha+1}}, E[X]=θα−1E[X]=\frac{\theta}{\alpha-1} (α>1\alpha>1), Var(X)=αθ2(α−1)2(α−2)\text{Var}(X)=\frac{\alpha\theta^2}{(\alpha-1)^2(\alpha-2)} (α>2\alpha>2); α = shape, θ = scale, x > 0
Finite-population correction factor for hypergeometric variance
Var(X)=n⋅KN⋅N−KN⋅N−nN−1\text{Var}(X) = n\cdot\frac{K}{N}\cdot\frac{N-K}{N}\cdot\frac{N-n}{N-1}, n = draws, N = population, K = successes in population, X = successes drawn
Sum of independent Poisson random variables
Poisson(λ1)+Poisson(λ2)=Poisson(λ1+λ2)\text{Poisson}(\lambda_1) + \text{Poisson}(\lambda_2) = \text{Poisson}(\lambda_1 + \lambda_2), λ₁, λ₂ = rates of independent Poisson counts; combined count is Poisson with summed rate
Discrete uniform distribution pmf mean and variance
P(X=x)=1k, E[X]=a+b2, Var(X)=k2−112P(X=x)=\tfrac{1}{k},\ E[X]=\tfrac{a+b}{2},\ \text{Var}(X)=\tfrac{k^2-1}{12}; a, b = min, max integers; k = b-a+1 = count of values; x in {a,...,b}
Gamma-Poisson tail connection for integer shape
P(X≤t)=P(Poisson(t/θ)≥α)P(X \le t) = P(\text{Poisson}(t/\theta) \ge \alpha), equivalently P(X>t)=P(Poisson(t/θ)≤α−1)P(X > t) = P(\text{Poisson}(t/\theta) \le \alpha - 1); X~Gamma(α,θ), α = integer shape, θ = scale, t = time
Beta distribution variance
Var(X)=ab(a+b)2(a+b+1)\text{Var}(X) = \frac{ab}{(a+b)^2(a+b+1)}, a, b = shape parameters of Beta on (0,1)
Rectangular probability from a joint CDF
P(a<X≤b, c<Y≤d)=F(b,d)−F(a,d)−F(b,c)+F(a,c)P(a < X \leq b,\ c < Y \leq d) = F(b,d) - F(a,d) - F(b,c) + F(a,c), F = joint CDF; a,b = X limits; c,d = Y limits
Conditional PMF of a discrete random variable
pY∣X(y∣x)=pX,Y(x,y)pX(x) p_{Y|X}(y|x) = \dfrac{p_{X,Y}(x,y)}{p_X(x)} , pX,Yp_{X,Y} = joint PMF, pXp_X = marginal PMF of X, requires pX(x)>0p_X(x) > 0
Compound mean (Wald's identity for the mean)
E[S]=E[N]⋅E[X]E[S] = E[N] \cdot E[X], S = aggregate loss, N = random claim count, X = i.i.d. severity independent of N
Variance of a compound random sum
Var(S)=E[N] Var(X)+Var(N) (E[X])2 \text{Var}(S) = E[N]\,\text{Var}(X) + \text{Var}(N)\,(E[X])^2 , S = sum of N losses, N = random claim count, X = individual loss severity
Covariance via conditional expectation
Cov(X,Y)=E[X⋅E[Y∣X]]−E[X]⋅E[Y] \text{Cov}(X,Y) = E[X \cdot E[Y|X]] - E[X] \cdot E[Y] , E[Y|X] = conditional mean of Y given X, E[X], E[Y] = marginal means
PDF of the k-th order statistic
fX(k)(x)=n!(k−1)!(n−k)![F(x)]k−1[1−F(x)]n−kf(x)f_{X_{(k)}}(x) = \frac{n!}{(k-1)!(n-k)!}[F(x)]^{k-1}[1-F(x)]^{n-k}f(x), n = sample size, k = rank, F = cdf, f = pdf
Linear combination of independent normal random variables
aX+bY+c∼N(aμX+bμY+c,  a2σX2+b2σY2)aX + bY + c \sim N(a\mu_X + b\mu_Y + c,\; a^2\sigma_X^2 + b^2\sigma_Y^2), a, b, c = constants, μ = mean, σ² = variance; X, Y independent normals
Mean of a linear combination of random variables
E[∑i=1naiXi+b]=∑i=1naiE[Xi]+bE\left[\sum_{i=1}^n a_i X_i + b\right] = \sum_{i=1}^n a_i E[X_i] + b, a_i = constant coefficients, X_i = random variables, b = additive constant (no independence needed)
Central Limit Theorem normal approximation for a sum
Sn=∑i=1nXi≈N(nμ,nσ2)S_n = \sum_{i=1}^n X_i \approx N(n\mu, n\sigma^2), standardize z=s−nμσnz = \frac{s - n\mu}{\sigma\sqrt{n}}; μ\mu = mean, σ2\sigma^2 = variance, n = number of iid terms

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