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SECTIONProbability32 subsectionsBROWSE ALL ↓INTERACTIVE2 TOOLSProbability VisualTools13 toolsJump to ↓ProbabilityCalculator Suite9 toolsJump to ↓fREFERENCE165 ITEMSfProbabilityFormulas Reference108 itemsJump to ↓AaDefinitions57 itemsJump to ↓§CORE TOPICS28 SUBSECTIONSProbability AxiomsJump to ↓Probability BasicsJump to ↓Bayes\Jump to ↓Cumulative DistributionFunction (CDF)Jump to ↓Central Limit TheoremJump to ↓Combinatorics in ProbabilityJump to ↓Conditional ProbabilityJump to ↓Covariance in ProbabilityJump to ↓Probability Distributions2 TOPICSJump to ↓Events in ProbabilityJump to ↓Expected ValueJump to ↓Independence of EventsJump to ↓Indicator Random VariablesJump to ↓Probability Inequalities2 TOPICSJump to ↓Joint ProbabilityJump to ↓Key TermsJump to ↓Key TermsJump to ↓Mode in ProbabilityJump to ↓Probability Models2 TOPICSJump to ↓Probability function2 TOPICSJump to ↓Random VariablesJump to ↓Rules of ProbabilityJump to ↓UncountableJump to ↓Sets in ProbabilityJump to ↓Law of Total ProbabilityJump to ↓Tree Diagrams in ProbabilityJump to ↓Variance in ProbabilityJump to ↓Probability SymbolsReferenceJump to ↓

Probability Visual Tools

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Visual probability tools transform abstract mathematical concepts into interactive experiences that build intuitive understanding. By manipulating parameters and observing real-time changes in distributions, sample spaces, and probability outcomes, you develop the kind of deep, geometric intuition that makes probability theory truly click — moving beyond memorized formulas to genuine comprehension of how randomness behaves.
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Cumulative Distribution Function Visualizers

Understand cumulative distribution functions through interactive charts for discrete (Binomial, Poisson, Geometric) and continuous distributions (Normal, Exponential, Uniform). Adjust parameters to see how probability accumulates across different distributions with real-time CDF visualizations.

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H0.5
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Coin Toss Probability Simulator and Calculator

Flip a fair or biased coin thousands of times and watch the Law of Large Numbers pull results toward theory, or map the complete sample space for up to six flips and calculate the probability of any pattern you highlight.

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P(A | B)
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Conditional Probability Visualization Tools

Explore conditional probability with interactive tree diagrams, Venn diagrams, waffle charts, and contingency tables. Visual tools for understanding P(A|B).

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nine distributions
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Probability Distribution Explorers

Explore discrete and continuous probability distributions with interactive calculators and visualizations. Calculate probabilities, understand parameters, and visualize distributions including binomial, normal, Poisson, and exponential.

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E[X]
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Expected Value Visualization Tools

See how probabilities pull the expected value toward heavier outcomes with a weight-and-force visualization, or adjust discrete distributions with sliders and watch E(X) shift in real time.

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tail bounds
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Probability Inequality Visualization Tools

Visualize the Markov and Chebyshev inequalities across nine distributions: adjust thresholds, compare actual tail probabilities to the theoretical bounds, and learn when each bound is tight.

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Probability Formulas Reference

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Reference of 108 probability formulas across 30 categories: axioms, conditional probability, expectation, variance, distributions, moment generating functions.
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Definitions

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Discrete Distributions7
Negative Binomial DistributionA discrete distribution counting the number of trials needed to achieve a fixed number rr of successes in a sequence of independent Bernoulli trials…Read more →Bernoulli DistributionA discrete distribution for a single trial with two outcomes: P(X=1)=pP(X = 1) = p and P(X=0)=1−pP(X = 0) = 1 - p.Read more →Binomial DistributionA discrete distribution counting the number of successes in nn independent Bernoulli trials, each with success probability pp.Read more →Poisson DistributionA discrete distribution modelling the number of events occurring in a fixed interval, where events happen independently at a constant average rate…Read more →Discrete Uniform DistributionA discrete distribution where each of nn possible values has equal probability 1/n1/n.Read more →Geometric DistributionA discrete distribution counting the number of trials needed to obtain the first success in a sequence of independent Bernoulli trials with success…Read more →Hypergeometric DistributionA discrete distribution describing the number of successes in nn draws without replacement from a finite population containing KK successes and…Read more →
ProbabilityA function PP that assigns to each event AA in a sample space a real number P(A)∈[0,1]P(A) \in [0, 1] satisfying the probability axioms.Read more →Random ExperimentA process or action whose outcome cannot be predicted with certainty before it is performed.Read more →Sample SpaceΩ={ω1,ω2,…}\Omega = \{\omega_1, \omega_2, \ldots\} — the set of all possible outcomes of a random experiment.Read more →EventA⊆ΩA \subseteq \Omega — a subset of the sample space.Read more →Elementary EventAn event consisting of exactly one outcome: {ω}\{\omega\} where ω∈Ω\omega \in \Omega.Read more →Relative Frequencyfn(A)=number of times A occursnf_n(A) = \frac{\text{number of times } A \text{ occurs}}{n} where nn is the total number of trials.Read more →Probability MeasureA function P:F→[0,1]P: \mathcal{F} \to [0,1] defined on a collection of events, satisfying non-negativity, normalization (P(Ω)=1P(\Omega) = 1), and countable…Read more →Equally Likely EventsEvents A1,A2,…,AnA_1, A_2, \ldots, A_n are equally likely when P(A1)=P(A2)=⋯=P(An)P(A_1) = P(A_2) = \cdots = P(A_n).Read more →
Expected ValueE[X]=∑xx⋅pX(x)E[X] = \sum_x x \cdot p_X(x) (discrete) or E[X]=∫−∞∞x⋅fX(x) dxE[X] = \int_{-\infty}^{\infty} x \cdot f_X(x)\,dx (continuous).Read more →VarianceVar⁡(X)=E[(X−μ)2]\operatorname{Var}(X) = E[(X - \mu)^2] where μ=E[X]\mu = E[X].Read more →Standard DeviationσX=Var⁡(X)\sigma_X = \sqrt{\operatorname{Var}(X)}Read more →CovarianceCov⁡(X,Y)=E[(X−E[X])(Y−E[Y])]\operatorname{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])]Read more →Correlation CoefficientρXY=Cov⁡(X,Y)σX⋅σY\rho_{XY} = \frac{\operatorname{Cov}(X, Y)}{\sigma_X \cdot \sigma_Y}, where σX,σY>0\sigma_X, \sigma_Y > 0.Read more →Conditional ExpectationE[X∣Y=y]=∑xx⋅P(X=x∣Y=y)E[X \mid Y = y] = \sum_x x \cdot P(X = x \mid Y = y) (discrete) or E[X∣Y=y]=∫x⋅fX∣Y(x∣y) dxE[X \mid Y = y] = \int x \cdot f_{X|Y}(x \mid y)\,dx (continuous).Read more →Conditional VarianceVar⁡(X∣Y=y)=E[(X−E[X∣Y=y])2∣Y=y]\operatorname{Var}(X \mid Y = y) = E[(X - E[X \mid Y = y])^2 \mid Y = y]Read more →Moment of a Random VariableThe kk-th moment of XX about the origin is E[Xk]E[X^k]. The kk-th central moment is E[(X−μ)k]E[(X - \mu)^k].Read more →
Bivariate Random VariableA pair of random variables (X,Y)(X, Y) defined on the same sample space, considered jointly.Read more →N-Variate Random VariablesA vector (X1,X2,…,Xn)(X_1, X_2, \ldots, X_n) of nn random variables defined on the same sample space.Read more →Independent Random VariablesRandom variables XX and YY are independent if P(X≤x,Y≤y)=P(X≤x)⋅P(Y≤y)P(X \le x, Y \le y) = P(X \le x) \cdot P(Y \le y) for all x,yx, y.Read more →Orthogonal Random VariablesRandom variables XX and YY are orthogonal if E[XY]=0E[XY] = 0.Read more →Uncorrelated Random VariablesRandom variables XX and YY are uncorrelated if Cov⁡(X,Y)=0\operatorname{Cov}(X, Y) = 0, equivalently E[XY]=E[X]E[Y]E[XY] = E[X]E[Y].Read more →Marginal DistributionThe distribution of one random variable obtained from a joint distribution by summing (discrete) or integrating (continuous) over all values of the…Read more →Joint Cumulative Distribution FunctionFX,Y(x,y)=P(X≤x,Y≤y)F_{X,Y}(x, y) = P(X \le x, Y \le y) for all x,y∈Rx, y \in \mathbb{R}.Read more →Joint Probability Mass FunctionpX,Y(x,y)=P(X=x,Y=y)p_{X,Y}(x, y) = P(X = x, Y = y) for discrete random variables XX and YY.Read more →Joint Probability Density FunctionA function fX,Y(x,y)≥0f_{X,Y}(x,y) \ge 0 such that P((X,Y)∈A)=∬AfX,Y(x,y) dx dyP((X,Y) \in A) = \iint_A f_{X,Y}(x,y)\,dx\,dy for any region AA.Read more →Conditional Probability Mass FunctionpX∣Y(x∣y)=pX,Y(x,y)pY(y)p_{X|Y}(x \mid y) = \frac{p_{X,Y}(x, y)}{p_Y(y)}, defined when pY(y)>0p_Y(y) > 0.Read more →Conditional Probability Density FunctionfX∣Y(x∣y)=fX,Y(x,y)fY(y)f_{X|Y}(x \mid y) = \frac{f_{X,Y}(x, y)}{f_Y(y)}, defined when fY(y)>0f_Y(y) > 0.Read more →
Venn DiagramA graphical representation using overlapping circles to depict sets (events) and their relationships within a sample space.Read more →Null Set∅\emptyset — the set containing no elements, representing an impossible event in probability.Read more →Union of SetsA∪B={ω:ω∈A or ω∈B}A \cup B = \{\omega : \omega \in A \text{ or } \omega \in B\} — the event that at least one of AA or BB occurs.Read more →Intersection of SetsA∩B={ω:ω∈A and ω∈B}A \cap B = \{\omega : \omega \in A \text{ and } \omega \in B\} — the event that both AA and BB occur simultaneously.Read more →Disjoint SetsSets AA and BB are disjoint if A∩B=∅A \cap B = \emptyset — they share no common elements.Read more →Complement of a SetAc={ω∈Ω:ω∉A}A^c = \{\omega \in \Omega : \omega \notin A\} — all outcomes in the sample space that are not in AA.Read more →
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Probability Axioms

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Explore the three fundamental probability axioms that form the foundation of probability theory. Learn non-negativity, normalization, and additivity with clear explanations.
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Probability Basics

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The complete conceptual tour of probability on one page: sample spaces and events, the Kolmogorov axioms, the full tree of probability rules, combinatorial counting, random variables and their distributions, conditional probability and independence, and the probability function that underlies it all — illustrated with diagrams and a comprehensive key-terms map into the definitions glossary.
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Bayes\

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Learn Bayes
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Cumulative Distribution Function (CDF)

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Learn cumulative distribution function (CDF) in probability. Understand how probability accumulates, CDF properties, and calculating probabilities for random variables.
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Central Limit Theorem

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Learn the Central Limit Theorem: how sample means converge to normal distribution. Understand CLT conditions, applications in statistical inference, and key differences from Law of Large Numbers.
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Combinatorics in Probability

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Learn how combinatorics and counting methods form the foundation of classical probability. Understand finite sample spaces, equally likely outcomes, and the classical probability formula.
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Conditional Probability

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Learn conditional probability: how probabilities update with new information. Understand the P(A|B) formula, restricted sample spaces, and the difference between conditioning and independence.
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Covariance in Probability

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Learn how covariance measures joint behavior of random variables. Understand positive, negative, and zero covariance, independence relationships, and applications.
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Probability Distributions

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Every outcome in a finite set has exactly the same probability—complete symmetry across the support.
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Events in Probability

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Learn probability events: how outcomes group into meaningful situations. Understand sample space, event operations, union, intersection, independence, and set-theoretic foundations.
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Expected Value

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The expected value E(X) as a weighted average of outcomes: definition, computation for discrete and continuous random variables, linearity properties, and worked examples building from simple bets to full distributions.
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Independence of Events

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Learn independence in probability: when events don
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Indicator Random Variables

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Learn indicator random variables: converting events to 0-1 random variables. Understand E[I_A] = P(A), linearity of expectation, counting with indicators, and applications.
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Probability Inequalities

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Learn probability inequalities: bounding probabilities with limited information. Understand Markov, Chebyshev, tail bounds, distribution-free bounds, and trade-offs between assumptions and tightness.
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Joint Probability

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Key Terms

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Key Terms

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Mode in Probability

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Learn how mode identifies peak probability concentration in distributions. Understand mode for discrete and continuous cases, multiple modes, and comparison with mean and median.
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Probability Models

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Learn probability models: mathematical descriptions of random situations. Understand sample spaces, events, probability assignments, models vs distributions, random variables, and modeling assumptions.
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Probability function

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Learn probability functions: PMF for discrete variables, PDF for continuous variables, properties, notation, determining probability functions, and connections to distributions and expectations.
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Random Variables

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Rules of Probability

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The working toolkit of probability: axiomatic bounds, complement and difference rules, addition and inclusion-exclusion, multiplication and chain rules, the law of total probability, and the Bayes theorem — each stated formally and illustrated.
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Uncountable

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Learn sample space in probability: definition, notation, types (finite, infinite, continuous), listing outcomes, properties, relationship to events, and practical examples.
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Sets in Probability

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Learn sets in probability: definition, notation, types, set operations (union, intersection, complement), how sets become events, role in probability models, and connections to other concepts.
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Law of Total Probability

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Learn the law of total probability: when to use, splitting probability across cases, formula, diagrams, examples, relationship to Bayes
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Tree Diagrams in Probability

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Learn probability tree diagrams: structure, components, computing probabilities through paths, conditional probability connection, law of total probability visualization, Bayes
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Variance in Probability

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Learn variance: definition, calculation methods, why square deviations, properties, variance of sums, standard distributions, standard deviation comparison, common mistakes, and connections to expectation and covariance.
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Probability Calculator Suite

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Update probabilities
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Probability Symbols Reference

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Our Probability Symbols page delivers a systematic reference for notation used in probability theory and statistics. This collection serves as an essential guide for students and professionals working with statistical concepts.

The reference organizes symbols into practical categories including probability notations (P(A), P(A|B)), random variables and distributions (fX(x), FX(x)), and common distribution families (Bin(n,p), N(μ,σ²)). It extends to advanced topics like statistical measures (E(X), Var(X)), hypothesis testing parameters (H₀, α, p-value), and information theory metrics (H(X), I(X;Y)).

Specialized sections cover moment generating functions (MX(t)), key probability inequalities (Markov’s, Chebyshev’s), Bayesian methods, and regression analysis notation — all presented with precise LaTeX formatting to support academic writing and research in probability and statistics.

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