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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 (A^{c}) = 1 - P (A)

Addition rule (two events)

P (A \cup B) = P (A) + P (B) - P (A \cap B)

Addition rule (three events)

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)

_{n} P_{r} = \frac{n !}{( n - r )!}

n

objects taken

r

at a time, order matters

Combinations (unordered)

(r n) = \frac{n !}{r ! ( n - r )!}

n

objects taken

r

at a time, order does not matter

Conditional Probability & Bayes 5 items

Conditional probability

P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}, P (B) > 0

Multiplication rule

P (A \cap B) = P (A ∣ B) P (B) = P (B ∣ A) P (A)

Law of total probability

P (A) = \sum_{i} P (A ∣ B_{i}) P (B_{i})

where

{B_{i}}

is a partition of the sample space

Bayes' theorem

P (B_{i} ∣ A) = \frac{P ( A ∣ B _{i} ) P ( B _{i} )}{\sum _{j} P ( A ∣ B _{j} ) P ( B _{j} )}

Independence of two events

A

and

B

are independent iff

P (A \cap B) = P (A) P (B)

(equivalently:

P (A ∣ B) = P (A)

)

Discrete Distributions 6 items

Bernoulli distribution — PMF, mean, variance

P (X = 1) = p, P (X = 0) = 1 - p

E [X] = p, Var (X) = p (1 - p)

Binomial distribution — PMF, mean, variance

P(X=k)=(kn​)pk(1−p)n−k
E[X]=np,Var(X)=np(1−p)
k=0,1,…,n

Poisson distribution — PMF, mean, variance

P(X=k)=k!e−λλk​
E[X]=λ,Var(X)=λ
k=0,1,2,…

Geometric distribution — PMF, mean, variance

P(X=k)=(1−p)k−1p (number of trials to first success)
E[X]=p1​,Var(X)=p21−p​
k=1,2,…

Negative Binomial distribution — PMF, mean, variance

P (X = k) = (r - 1 k - 1) p^{r} (1 - p)^{k - r}

(trials for

r

th success)

E [X] = \frac{r}{p}, Var (X) = \frac{r ( 1 - p )}{p ^{2}}

Hypergeometric distribution — PMF, mean, variance

P (X = k) = \frac{( k K ) ( n - k N - K )}{( n N )}

E [X] = \frac{n K}{N}, Var (X) = \frac{n K ( N - K ) ( N - n )}{N ^{2} ( N - 1 )}

Continuous Distributions 8 items

Uniform distribution — PDF, mean, variance

f (x) = \frac{1}{b - a}, a < x < b

E [X] = \frac{a + b}{2}, Var (X) = \frac{( b - a ) ^{2}}{12}

Exponential distribution — PDF, CDF, mean, variance

f (x) = λ e^{- λ x}, F (x) = 1 - e^{- λ x}, x > 0

E [X] = \frac{1}{λ}, Var (X) = \frac{1}{λ ^{2}}

Memoryless property (Exponential / Geometric)

P (X > s + t ∣ X > s) = P (X > t)

Only the Exponential (continuous) and Geometric (discrete) satisfy this.

Normal distribution — PDF

f (x) = \frac{1}{σ 2 π} exp (- \frac{( x - μ ) ^{2}}{2 σ ^{2}})

E [X] = μ, Var (X) = σ^{2}

Gamma distribution — PDF, mean, variance

f(x)=Γ(α)θαxα−1e−x/θ​,x>0
E[X]=αθ,Var(X)=αθ2
α=shape, θ=scale

Weibull distribution — PDF, mean

f (x) = \frac{τ}{θ} (\frac{x}{θ})^{τ - 1} e^{- (x / θ)^{τ}}, x > 0

E [X] = θ Γ (1 + \frac{1}{τ})

Lognormal distribution — PDF, mean, variance

f (x) = \frac{1}{x σ 2 π} exp (- \frac{( ln x - μ ) ^{2}}{2 σ ^{2}}), x > 0

E [X] = e^{μ + σ^{2} /2}, Var (X) = e^{2 μ + σ^{2}} (e^{σ^{2}} - 1)

Beta distribution — PDF, mean

f (x) = \frac{x ^{α - 1} ( 1 - x ) ^{β - 1}}{B ( α , β )}, 0 < x < 1

E [X] = \frac{α}{α + β}, Var (X) = \frac{α β}{( α + β ) ^{2} ( α + β + 1 )}

Joint & Marginal Distributions 3 items

Joint PDF — marginal densities

f_{X} (x) = \int_{- \infty}^{\infty} f_{X, Y} (x, y) d y

f_{Y} (y) = \int_{- \infty}^{\infty} f_{X, Y} (x, y) d x

Conditional PDF

f_{Y ∣ X} (y ∣ x) = \frac{f _{X, Y} ( x , y )}{f _{X} ( x )}

Independence of continuous RVs

X, Y

independent iff

f_{X, Y} (x, y) = f_{X} (x) f_{Y} (y)

for all

x, y

Expectation & Variance 7 items

MGF definition and moment extraction

M_{X} (t) = E [e^{tX}]

E [X^{n}] = M_{X}^{(n)} (0)

(

n

th derivative at

t = 0

)

Variance shortcut

Var (X) = E [X^{2}] - (E [X])^{2}

Covariance

Cov (X, Y) = E [X Y] - E [X] E [Y]

X, Y

independent:

Cov (X, Y) = 0

Correlation coefficient

ρ_{X Y} = \frac{Cov ( X , Y )}{σ _{X} σ _{Y}}, - 1 \leq ρ \leq 1

Variance of a linear combination

Var (a X + bY) = a^{2} Var (X) + b^{2} Var (Y) + 2 ab Cov (X, Y)

Law of total expectation

E [X] = E [E [X ∣ Y]]

Law of total variance

Var (X) = E [Var (X ∣ Y)] + Var (E [X ∣ Y])

Insurance Applications 5 items

Ordinary deductible — payment per loss

YL=max(X−d,0)
E[YL]=E[X]−E[X∧d]
where d = deductible, X = ground-up loss

Limited expected value (LEV)

E [X \land u] = \int_{0}^{u} S (x) d x = \int_{0}^{u} [1 - F (x)] d x

for

X \geq 0

Payment per payment (excess loss)

e (d) = E [X - d ∣ X > d] = \frac{E [ X ] - E [ X \land d ]}{1 - F ( d )}

d

= deductible

Policy limit — payment per loss

YL=min(X,u)=X∧u
E[YL]=E[X∧u]
u = policy limit

Stop-loss (aggregate) premium

E [(S - d)_{+}] = E [S] - E [S \land d]

S

= aggregate loss,

d

= retention (aggregate deductible)

Topic 0 28 items

De Morgan's Laws

(A \cup B)^{c} = A^{c} \cap B^{c}

and

(A \cap B)^{c} = A^{c} \cup B^{c}

; A, B = events,

^{c}

= complement,

\cup

= union,

\cap

= intersection

Stars and bars distribution count

(k - 1 n + k - 1)

, n = identical objects distributed, k = distinct bins (non-negative integer solutions to

x_{1} + \dots + x_{k} = n

); use

(k - 1 n - 1)

if each bin needs at least one

Addition rule for mutually exclusive events

P (A \cup B) = P (A) + P (B)

when

P (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) \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 (A_{1} \cap \dots \cap A_{n}) = P (A_{1}) P (A_{2} ∣ A_{1}) P (A_{3} ∣ A_{1} \cap A_{2}) \dots P (A_{n} ∣ A_{1} \cap \dots \cap A_{n - 1})

, each factor conditions on all prior events

Odds form of Bayes' theorem

\frac{P ( B ∣ A )}{P ( B ^{c} ∣ A )} = \frac{P ( A ∣ B )}{P ( A ∣ 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 = \frac{n !}{n _{1} ! n _{2} ! \dots 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

S_{X} (x) = P (X > x) = 1 - F_{X} (x)

S_{X}

= survival function,

F_{X} (x) = P (X \leq x)

= CDF, x = threshold value

Conditional CDF of a left-truncated distribution

F_{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)] = \int_{- \infty}^{\infty} g (x) f_{X} (x) d x

, g = function of X, f_X = pdf of X (use sum with p_X(x) if discrete)

Chebyshev's inequality

P (∣ X - μ ∣ \geq k σ) \leq \frac{1}{k ^{2}}

, μ = mean, σ = standard deviation, k = number of SDs; equivalently at least

1 - 1/ k^{2}

lies within k SDs

Expected payment per loss with deductible and benefit limit

E [Y^{L}] = E [X \land (d + u)] - E [X \land d]

, d = deductible, u = benefit limit, X = ground-up loss,

X \land a

= loss capped at a

Second moment of the excess loss for an exponential

E [((X - d)_{+})^{2}] = S (d) \cdot 2 θ^{2} = e^{- d / θ} \cdot 2 θ^{2}

, d = deductible, θ = exponential mean, S(d) = survival function at d

Pareto distribution pdf, mean, and variance

f (x) = \frac{α θ ^{α}}{( x + θ ) ^{α + 1}}

E [X] = \frac{θ}{α - 1}

(

α > 1

Var (X) = \frac{α θ ^{2}}{( α - 1 ) ^{2} ( α - 2 )}

(

α > 2

); α = shape, θ = scale, x > 0

Finite-population correction factor for hypergeometric variance

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 (λ_{1}) + Poisson (λ_{2}) = Poisson (λ_{1} + λ_{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) = \frac{1}{k}, E [X] = \frac{a + b}{2}, Var (X) = \frac{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 \leq t) = P (Poisson (t / θ) \geq α)

, equivalently

P (X > t) = P (Poisson (t / θ) \leq α - 1)

; X~Gamma(α,θ), α = integer shape, θ = scale, t = time

Beta distribution variance

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 \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

p_{Y ∣ X} (y ∣ x) = \frac{p _{X, Y} ( x , y )}{p _{X} ( x )}

p_{X, Y}

= joint PMF,

p_{X}

= marginal PMF of X, requires

p_{X} (x) > 0

Compound mean (Wald's identity for the mean)

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}

, S = sum of N losses, N = random claim count, X = individual loss severity

Covariance via conditional expectation

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

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

a X + bY + c \sim N (a μ_{X} + b μ_{Y} + c, a^{2} σ_{X}^{2} + b^{2} σ_{Y}^{2})

, a, b, c = constants, μ = mean, σ² = variance; X, Y independent normals

Mean of a linear combination of random variables

E [\sum_{i = 1}^{n} a_{i} X_{i} + b] = \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

S_{n} = \sum_{i = 1}^{n} X_{i} \approx N (n μ, n σ^{2})

, standardize

z = \frac{s - n μ}{σ n}

;

μ

= mean,

σ^{2}

= variance, n = number of iid terms

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