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. 40 formulas across 7 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

40 Formulas
7 Topics
2026 Syllabus
Free Forever
Print-ready PDF: 1080x1350 portrait, math pre-rendered, fonts embedded. Download once, study anywhere.
Download PDF →

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 9 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
Pareto distribution — PDF, mean, variance
f(x)=α θα(x+θ)α+1,x>0f(x)=\dfrac{\alpha\,\theta^\alpha}{(x+\theta)^{\alpha+1}},\quad x>0
E[X]=θα−1  (α>1),Var(X)=αθ2(α−1)2(α−2)  (α>2)E[X]=\dfrac{\theta}{\alpha-1}\;(\alpha>1),\quad \text{Var}(X)=\dfrac{\alpha\theta^2}{(\alpha-1)^2(\alpha-2)}\;(\alpha>2)
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)

Frequently Asked Questions

Is the Exam P formula sheet free?
Yes. The full Exam P formula sheet is free, with no signup, no email, and no credit card required. 40 formulas across 7 topics, all rendered with the same KaTeX math notation used in the FreeFellow study app.
Can I download the Exam P formula sheet as a printable PDF?
Yes. A 1080x1350 portrait PDF (Instagram and LinkedIn carousel native size, also great for tablet study) is linked at the top of this page. The PDF is fully self-contained: math is pre-rendered, fonts are embedded, no internet connection needed once downloaded.
What's covered on the Exam P formula sheet?
Every formula is grouped by official syllabus topic, with the formula in math notation plus a one-line note on when to use it (or a watch-out from CAIA, CFA, or other prep-provider commentary). Coverage is calibrated to the 2026 syllabus and refreshed when the corpus changes.
Is this formula sheet affiliated with SOA?
No. FreeFellow is not affiliated with the SOA or any examination body. This is an independent study aid covering the published syllabus.
What else is free at FreeFellow for Exam P candidates?
The full question bank with detailed solutions, mixed practice, readiness tracking, lessons (where available), and the formula sheet are all free forever. Fellow ($59/quarter or $149/year per track) unlocks timed mock exams, spaced-repetition flashcards, performance analytics, AI essay grading, and a personalized study plan.
Practice Exam P questions free →

Formula sheets for other exams

Exam FM Exam FAM Exam SRM Exam ALTAM Exam ASTAM CFA Level I CFA Level II CFA L3 Portfolio Mgmt CFA L3 Private Wealth CFA L3 Private Markets CFP CPA AUD

About FreeFellow

FreeFellow is an AI-native exam prep platform for actuarial (SOA & CAS), CFA, CFP, CPA, CAIA, and securities licensing candidates — built around modern AI as a core capability rather than as a bolt-on. Every lesson ships with AI-narrated audio. Every constructed-response item has a copy-to-AI prompt builder so candidates can paste their answer into their own ChatGPT or Claude for self-graded feedback. Fellow members get instant AI grading on essays against the official rubric (currently CFA Level III, expanding to other essay-bearing sections).

The 70% you need to pass — question bank, written solutions, lessons, formula sheet, mixed practice, readiness tracking — is free forever, with no trial period and no credit card. Become a Fellow ($59/quarter or $149/year per track) to unlock mock exams, flashcards with spaced repetition, performance analytics, AI essay grading, and a personalized study plan.