What does this Probability cheat sheet cover?

It has 11 sections: Counting, Axioms, Conditional, Independence, Bayes, Discrete RVs, Common Distributions, Joint Laws, Expectation, Inequalities and Limit Laws. Together they cover the topics a typical Probability exam tests, including counting: permutations and combinations, axioms and conditional probability, independence and Bayes' theorem.

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The first part is free to read on this page, with no account needed. The full sheet opens in the app with Unlimited ($4.99 a month or $19 for a semester), where you can edit it, print it or save it as a PDF. It is one US Letter page and prints cleanly in black and white or colour.

Can I bring this Probability cheat sheet to my exam?

Only if your course allows a note sheet, so check the syllabus. This one is a single US Letter page. If your exam allows a different size, such as a 3x5 or 4x6 index card or one page front and back, Cheat Sheet AI makes one at exactly that size from your own lecture notes.

The sheet was written by our engine from a standard course outline, then a second model fact-checked every formula and statement against it; it corrected 3 statements before publishing. Notation and emphasis vary between courses, so check it against your own lecture notes.

Can I make a Probability sheet from my own notes?

Yes. Upload your lecture slides, notes or study guide and Cheat Sheet AI writes a sheet in your course's notation, sized to exactly the space your exam allows. Your first sheets are free.

MathProbability

Probability Cheat Sheet — 1-page PDF

Counting, probability rules, random variables, common distributions, joint laws, inequalities, LLN, CLT. The whole course on one dense, printable page for your midterm or final, with every formula typeset and worked examples where they help.

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For a specific exam: Probability Final exam study sheet.

1 pageUS Letter
11 sections11 topics
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Preview of the one-page Probability cheat sheet: Counting, probability rules, random variables, common distributions, joint laws, inequalities, LLN, CLT — the whole course, one page

On the sheet

What’s on it

The topics a typical Probability exam tests, all on the one page.

Counting: permutations and combinations
Axioms and conditional probability
Independence
Bayes' theorem
Discrete random variables, expectation, variance
Common discrete distributions
Common continuous distributions
Joint distributions, covariance
Linearity of expectation
Markov and Chebyshev inequalities
Law of large numbers, CLT

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The free part of the sheet, section by section. Unlock the full sheet to read the rest, edit any block, and print it or save it as a PDF.

01Counting

Permutation: Ordered arrangement of $r$ objects chosen from $n$ distinct objects.

Permutations

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

Ordered selection, no repetition; $0 \leq r \leq n$ .

Combination: Unordered selection of $r$ objects from $n$ distinct objects.

Combinations

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

Choose $r$ from $n$ ; order irrelevant.

Binomial identity

(r n) = (n - r n)

Symmetry of choose.

Counting rules

Situation	Count
$r$ positions, repetition allowed	$n^{r}$
Arrange all $n$ distinct	$n!$
Circular order	$(n - 1)!$
Choose $r$ then order	$(r n) r! =_{n} P_{r}$

Ex.How many 3-card hands from 52 cards?

Unordered selection: $(3 52)$ .

Watch out:Order matters for permutations, not for combinations; do not divide by $r!$ unless all orderings of each set are equally counted.

02Conditional

Ex.If $P (A \cap B) = 0.12$ and $P (B) = 0.2$ , compute $P (A ∣ B)$ .

$P (A ∣ B) = 0.12/0.2 = 0.6$ .

Watch out: $P (A ∣ B)$ undefined if $P (B) = 0$ ; conditioning changes the probability model, not just a ratio.

03Independence

Independence: $A$ and $B$ independent iff $P (A \cap B) = P (A) P (B)$ .

Equivalent form

P (A ∣ B) = P (A)

Requires $P (B) > 0$ .

Mutual independence: $A_{1}, \dots, A_{n}$ mutually independent iff for every nonempty subset $I$ , $P (\cap_{i \in I} A_{i}) = \prod_{i \in I} P (A_{i})$ .

Independence facts

Statement	Also true
$A$ indep $B$	$A^{c}$ indep $B$
$A$ indep $B$	$A$ indep $B^{c}$
$A$ indep $B$	$A^{c}$ indep $B^{c}$

Watch out:Pairwise independence $\neq \Rightarrow$ mutual independence.

Ex.If $P (A) = 0.4$ , $P (B) = 0.5$ , independent, find $P (A \cap B)$ .

$0.4 \cdot 0.5 = 0.2$ .

04Common Distributions

Discrete distributions

Dist	PMF / params	Mean	Var
Bernoulli $(p)$	$P (X = 1) = p, P (X = 0) = 1 - p$	$p$	$p (1 - p)$
Binomial $(n, p)$	$P (X = k) = (k n) p^{k} (1 - p)^{n - k}$ , $k = 0, \dots, n$	$n p$	$n p (1 - p)$
Geometric $(p)$	$P (X = k) = (1 - p)^{k - 1} p$ , $k \geq 1$	$1/ p$	$(1 - p) / p^{2}$
Poisson $(λ)$	$P (X = k) = e^{- λ} λ^{k} / k!$ , $k = 0, 1, 2, \dots$	$λ$	$λ$

Binomial setup: $n$ independent Bernoulli trials, success prob. $p$ , count successes.
Geometric setup: Number of trials until first success; memoryless.

Poisson additivity

X_{i} \sim Pois (λ_{i}) indep \Rightarrow \sum_{i} X_{i} \sim Pois (\sum_{i} λ_{i})

Continuous distributions

Dist	pdf	Mean	Var
Uniform $(a, b)$	$1/ (b - a)$ on $[a, b]$	$(a + b) /2$	$(b - a)^{2} /12$
Exponential $(λ)$	$λ e^{- λ x}$ , $x \geq 0$ , $λ > 0$	$1/ λ$	$1/ λ^{2}$
Normal $(μ, σ^{2})$	$ϕ_{μ, σ} (x)$	$μ$	$σ^{2}$

05Expectation

Indicator

E [1_{A}] = P (A)

Key counting trick.

Variance rules

Var (a X + b) = a^{2} Var (X)

Shift does not change variance.

Sum variance

Var (\sum_{i} X_{i}) = \sum_{i} Var (X_{i}) + 2 \sum_{i < j} Cov (X_{i}, X_{j})

If independent, covariances $= 0$ .

Product expectation

X ⊥ Y \Rightarrow E [X Y] = E [X] E [Y]

Not conversely in general.

Ex.Let $X = \sum_{i = 1}^{10} I_{i}$ where $I_{i}$ indicates success with $P (I_{i} = 1) = p$ . Find $E [X]$ .

By linearity, $E [X] = \sum E [I_{i}] = 10 p$ .

Watch out:Variance is not linear: $Var (X + Y) = Var (X) + Var (Y)$ only if $X, Y$ independent (or at least uncorrelated).

06Inequalities

Markov

P (X \geq a) \leq \frac{E [ X ]}{a}

For $X \geq 0$ , $a > 0$ .

Chebyshev

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

$μ = E X$ , $σ^{2} = Var (X)$ , $k > 0$ .

General Chebyshev

P (∣ X - E X ∣ \geq c) \leq \frac{Var ( X )}{c ^{2}}

Equivalent form.

Cheat sheet locked

Also on the full Probability sheet

9 more sections, with every definition, formula and worked example. Unlimited unlocks every full sheet in the library to edit, print or save as a PDF, and makes unlimited sheets from your own notes.

Axioms

Sample space
Event
Axioms
Complement
Union
Difference
De Morgan

Conditional

Conditional prob
Multiplication
Chain rule
Partition
Total probability

Bayes

Bayes
Two-event Bayes
Prior / posterior

Discrete RVs

Random variable
PMF
CDF
Expectation
Variance
Std dev
Moment generating fn

Common Distributions

Normal pdf

Joint Laws

Joint pmf/pdf
Marginals
Conditional pmf/pdf
Covariance
Correlation
Independence

Expectation

Linearity

Inequalities

Union bound

Limit Laws

LLN
WLLN
CLT
Sum form
Use theorems when

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FAQ

Questions

About the Probability sheet and making your own.

What does this Probability cheat sheet cover?: It has 11 sections: Counting, Axioms, Conditional, Independence, Bayes, Discrete RVs, Common Distributions, Joint Laws, Expectation, Inequalities and Limit Laws. Together they cover the topics a typical Probability exam tests, including counting: permutations and combinations, axioms and conditional probability, independence and Bayes' theorem.
Is the probability cheat sheet free?: The first part is free to read on this page, with no account needed. The full sheet opens in the app with Unlimited ($4.99 a month or $19 for a semester), where you can edit it, print it or save it as a PDF. It is one US Letter page and prints cleanly in black and white or colour.
Can I bring this Probability cheat sheet to my exam?: Only if your course allows a note sheet, so check the syllabus. This one is a single US Letter page. If your exam allows a different size, such as a 3x5 or 4x6 index card or one page front and back, Cheat Sheet AI makes one at exactly that size from your own lecture notes.
How accurate is it?: The sheet was written by our engine from a standard course outline, then a second model fact-checked every formula and statement against it; it corrected 3 statements before publishing. Notation and emphasis vary between courses, so check it against your own lecture notes.
Can I make a Probability sheet from my own notes?: Yes. Upload your lecture slides, notes or study guide and Cheat Sheet AI writes a sheet in your course's notation, sized to exactly the space your exam allows. Your first sheets are free.

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