visual mathematics of uncertainty

Probability Atlas

A visual, interactive guide to the science of uncertainty: from sample spaces and Bayes' theorem to distributions, limit laws, Monte Carlo methods and stochastic processes.

Begin the journeyOpen distribution lab
0 → 1The complete probability scale
6 labsInteractive experiments
3 layersIntuition, formulas, simulation
∞ trialsThe ideal behind frequency
01 / Orientation

The knowledge map

Probability becomes easier when each topic is seen as part of one connected system. Counting describes possible worlds; probability weights them; random variables translate outcomes into numbers; distributions describe those numbers; limit laws explain stable patterns.

01

Model uncertainty

Experiments, outcomes, events and probability axioms.

 02

Count possibilities

Permutations, combinations and Pascal's triangle.

 03

Update knowledge

Conditioning, independence and Bayesian reversal.

 04

Measure outcomes

Random variables, PMFs, PDFs and CDFs.

 05

Summarize behavior

Expectation, variance, covariance and correlation.

 06

Reveal stability

Law of Large Numbers and Central Limit Theorem.

 07

Compute by chance

Simulation and Monte Carlo approximation.

 08

Follow random evolution

Random walks, Markov chains and long-run behavior.

 09

Build rigor

Measure theory, convergence, martingales and Brownian motion.

 10

Use the toolkit

A compact reference for the most important formulas.
02 / Foundations

Events, sets and the probability model

A probability model is a disciplined description of uncertainty. It identifies what can happen, which collections of outcomes matter, and how much probability each event receives.

The three-part model

Ω

Sample space: the set of all possible outcomes. For two coin tosses: Ω = {HH, HT, TH, TT}.

F

Events: selected subsets of Ω. Example: “exactly one head” is {HT, TH}.

P

Probability measure: a rule assigning each event a number from 0 to 1.

Probability space = (Ω, F, P)

Interactive event geometry

Choose a set operation and see which region is selected.

Venn lab
A B Ω
Union: outcomes in A, in B, or in both.
Axiom 1

Non-negativity

P(A) ≥ 0

Probability cannot be negative.

Axiom 2

Normalization

P(Ω) = 1

Some outcome in the model must occur.

Axiom 3

Countable additivity

P(⋃ Aᵢ) = Σ P(Aᵢ)

When the events are pairwise disjoint.

03 / Combinatorics

Count before you calculate

In finite equally likely models, probability is a ratio of counts. The central question becomes: how many favorable outcomes exist, and how many outcomes exist in total?

Permutation & combination calculator

Order matters for permutations; it does not matter for combinations.

Counting lab
10
3
Ordered selections
720
P(n,r)=n!/(n-r)!
Unordered selections
120
C(n,r)=n!/[r!(n-r)!]

Pascal's triangle

Each value is the sum of the two values above it and equals a binomial coefficient.

n choose r
11
0.50
path
Selected coefficientC(5,2) = 10
Row sum2^5 = 32
Binomial probability31.25%
Click or hover a cell to inspect C(n,r). The bars show P(X=r) for the selected row using p.
C(n,r) = C(n-1,r-1) + C(n-1,r)
C(5,2) = 5! / (2! 3!)

Falling balls board

Drop balls through a triangular peg board and watch the live histogram grow toward Pascal's binomial pattern.

Galton game
Click or tap above the pegs to choose the launch position.
Total landed0
Active balls0
Score0
Target bin-
Hottest bin-
Shape gap0.00%

Top 10 hit bins

    Hit the glowing target bin to build a streak. Adjacent bins still score partial points.
    First landing 100 balls 3-hit streak 1,000 points Shape gap under 8%

    Multiplication principle

    If one stage has m possibilities and the next has n, together they have mn.

    Permutations

    Use when the sequence or assignment matters: rankings, passwords without repetition, schedules.

    Combinations

    Use when only membership matters: committees, subsets, lottery selections.

    04 / Conditional probability

    Probability after information arrives

    Conditioning restricts the universe to outcomes compatible with new evidence. Bayes' theorem then reverses the direction of a conditional probability.

    The logic of conditioning

    P(A | B) = P(A ∩ B) / P(B)

    Read this as: among outcomes where B happened, what fraction also belong to A?

    Bayes' theorem

    P(A | B) = P(B | A)P(A) / P(B)

    The prior P(A) is updated by the likelihood P(B|A), producing the posterior P(A|B).

    Independence test
    A ⫫ B ⇔ P(A ∩ B)=P(A)P(B)

    Bayesian diagnostic simulator

    See why a highly accurate test can still produce surprising posterior probabilities.

    Base-rate lab
    5%
    95%
    90%
    true positivetrue negativefalse positivefalse negative
    Given a positive result, the probability of the condition is 33.3%
    05 / Random variables

    Turn outcomes into numbers

    A random variable is a function from outcomes to numerical values. This translation lets algebra and calculus operate on uncertainty.

    Discrete

    Probability mass function

    p(x)=P(X=x)

    Probability is concentrated on separate values. The masses sum to one.

    Continuous

    Probability density function

    P(a≤X≤b)=∫ₐᵇ f(x)dx

    Probability is area under a density curve. A single exact point usually has probability zero.

    Universal

    Cumulative distribution

    F(x)=P(X≤x)

    The CDF works for discrete, continuous and mixed distributions and never decreases.

    Outcome ω

    One possible result of the experiment.

    Variable X(ω)

    A numerical feature extracted from that outcome.

    Distribution

    How probability is allocated across values of X.

    Expectation

    The probability-weighted center.

    Variance

    The expected squared distance from the center.

    Inference

    Use observations to learn about hidden structure.

    06 / Distribution laboratory

    Explore the shapes of chance

    Different mechanisms generate different distribution families. Adjust parameters and watch the location, spread, skew and discreteness change.

    Mean
    Variance
    Type

    Bars represent probability mass; curves represent probability density. In a density, height is not probability by itself—area is.

    07 / Expectation and spread

    Center, variability and relationship

    Expectation summarizes long-run balance. Variance measures dispersion. Covariance and correlation describe how two random variables move together.

    Discrete mean
    E[X]=Σ x·p(x)

    A weighted average over possible values.

    Continuous mean
    E[X]=∫ x f(x)dx

    The density-weighted center of mass.

    Variance
    Var(X)=E[(X-μ)²]

    Average squared deviation from the mean.

    Correlation
    ρ=Cov(X,Y)/(σₓσᵧ)

    Standardized linear association from -1 to 1.

    Weighted balance: a loaded die

    Change the probability of rolling six; the remaining outcomes share the rest equally.

    Expectation lab
    0.17
    Expected value
    Variance
    Std. deviation

    Three ideas that prevent common mistakes

    1

    Expectation need not be a possible outcome. A fair die has mean 3.5, though 3.5 cannot be rolled.

    2

    Linearity is powerful. E[X+Y]=E[X]+E[Y], even without independence.

    3

    Zero correlation is weaker than independence. Nonlinear dependence can remain invisible to correlation.

    08 / Limit laws

    How randomness becomes regular

    The Law of Large Numbers stabilizes averages. The Central Limit Theorem explains why sums and averages often have an approximately normal shape.

    Law of Large Numbers

    Watch the running proportion of heads settle near the true probability.

    Frequency lab
    0.50
    1500
    Running proportion will appear here.

    Central Limit Theorem

    Average several dice repeatedly. As the number of dice grows, the histogram becomes bell-shaped.

    CLT lab
    2
    Increase the number of dice to see the normal shape emerge.

    Law of Large Numbers

    (X₁+⋯+Xₙ)/n → μ

    It is about convergence of the sample average to the population mean.

    Central Limit Theorem

    (Sₙ-nμ)/(σ√n) ⇒ N(0,1)

    It is about the limiting shape of standardized sums.

    09 / Simulation

    Monte Carlo: computation by random experiment

    Simulation turns probability into a numerical laboratory. It can estimate quantities that are difficult to compute analytically and reveal behavior before a proof is available.

    Estimate π with random points

    The fraction landing inside a quarter-circle approaches π/4.

    Monte Carlo lab
    3000
    π estimate: —

    The simulation cycle

    1

    Model the experiment and its assumptions.

    2

    Generate pseudorandom outcomes from the model.

    3

    Measure the quantity of interest in each run.

    4

    Aggregate results into an estimator.

    5

    Assess error by repetition, variance or confidence bounds.

    Typical Monte Carlo error ∝ 1/√n

    To reduce random error by a factor of 10, roughly 100 times as many trials are needed.

    10 / Stochastic processes

    Probability through time

    A stochastic process is a collection of random variables indexed by time or space. It models evolving uncertainty: queues, reliability, diffusion, finance, communications and many other systems.

    One-dimensional random walk

    At each step move +1 with probability p and -1 otherwise.

    Path lab
    0.50
    300
    Final position and extremes will appear here.

    Two-state Markov chain

    The next state depends only on the current state, not the complete past.

    State lab
    0.80
    0.30

    Beyond the introductory layer

    Advanced probability provides the rigorous language and tools needed for infinite sample spaces, dependent processes and continuous-time motion.

    σ-algebrasLebesgue integrationModes of convergenceBorel-Cantelli lemmasCharacteristic functionsMartingalesStopping timesErgodic theoremsPoisson processesBrownian motionItô calculusRenewal theory
    11 / Reference

    Formula field guide

    Use formulas as compressed statements of ideas. Before substituting numbers, identify the event, the assumptions and the meaning of each symbol.

    Complement

    P(Aᶜ)=1-P(A)

    Often the fastest route to “at least one.”

    Union of two events

    P(A∪B)=P(A)+P(B)-P(A∩B)

    Subtract overlap once.

    Total probability

    P(B)=ΣᵢP(B|Aᵢ)P(Aᵢ)

    Partition the sample space into cases.

    Bayes' theorem

    P(Aⱼ|B)=P(B|Aⱼ)P(Aⱼ)/ΣᵢP(B|Aᵢ)P(Aᵢ)

    Reverse a conditional probability.

    Binomial PMF

    P(X=k)=C(n,k)pᵏ(1-p)ⁿ⁻ᵏ

    Number of successes in n independent Bernoulli trials.

    Poisson PMF

    P(X=k)=e⁻λ λᵏ/k!

    Counts in a fixed interval under a constant rate model.

    Normal density

    f(x)=exp[-(x-μ)²/(2σ²)]/(σ√(2π))

    Symmetric continuous model with mean μ and variance σ².

    Exponential density

    f(x)=λe⁻λˣ, x≥0

    Waiting time under a constant event rate.

    Standardization

    Z=(X-μ)/σ

    Express a value in standard deviation units.

    Variance shortcut

    Var(X)=E[X²]-E[X]²

    Frequently simplifies calculation.

    Covariance of sums

    Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)

    The covariance term disappears under independence.

    Conditional expectation

    E[X]=E[E[X|Y]]

    The law of total expectation.

    12 / Study architecture

    A practical route through probability

    Move from intuition to calculation, then from calculation to proof. Simulation should accompany every stage rather than being postponed until the end.

    Level 1 · Foundation

    Learn the language

    1. Sets, events and sample spaces
    2. Counting principles
    3. Probability axioms
    4. Conditional probability and Bayes
    5. Independence
    Level 2 · Core calculus

    Learn the machinery

    1. Discrete and continuous variables
    2. PMF, PDF and CDF
    3. Expectation and variance
    4. Joint distributions and covariance
    5. Common distribution families
    Level 3 · Structure

    Learn the deep patterns

    1. Convergence concepts
    2. Law of Large Numbers
    3. Central Limit Theorem
    4. Markov chains and random walks
    5. Measure theory and stochastic processes

    Check 1

    A fair coin is tossed 10 times. Is the sequence HHHHHTTTTT more probable than HHTHTTHTHT?

    No. Each exact sequence has probability (1/2)¹⁰. What differs is the probability of a category of sequences, such as “exactly five heads.”

    Check 2

    Can two events be mutually exclusive and independent when both have positive probability?

    No. Mutually exclusive events have P(A∩B)=0, while independence would require P(A∩B)=P(A)P(B)>0.

    Check 3

    Does the Central Limit Theorem say that the original population becomes normal?

    No. It concerns the standardized distribution of sums or averages under suitable conditions.

    Check 4

    Why can a 95%-sensitive test still have a low probability of disease after a positive result?

    Because posterior probability also depends on prevalence and false-positive rate. A low base rate can make false positives outnumber true positives.
    13 / Study materials

    Sources used to shape this atlas

    The structure combines an intuitive introductory route with the deeper measure-theoretic and stochastic-process perspective found in advanced probability texts.

    Grinstead and Snell's Introduction to ProbabilitySimulation-first intuition, discrete and continuous models, limit laws, Markov chains and random walks.
    Probability: Theory and Examples — Rick DurrettMeasure-theoretic foundations, convergence, martingales, Markov chains, ergodic theory and Brownian motion.
    Probability and Statistics: The Science of Uncertainty — Evans & RosenthalProbability models, distributions, expectation, simulation, inference and stochastic processes.
    Probability Theory Lecture Notes — Phanuel MarianoCompact progression from combinatorics and axioms to distributions, covariance and limit laws.
    A First Course in Probability — Sheldon RossClassical problem-solving route through counting, conditional probability, random variables, limit theorems and simulation.
    Basic Probability Theory — Sharon GoldwaterAccessible explanations of notation, events, conditioning, random variables, expectation and estimation from data.