---
title: "Probability"
description: "37 topics in Mathematics, in the order they build on each other."
canonical: https://lightmysky.com/learn/mathematics/areas/probability
source: https://lightmysky.com/learn/mathematics/areas/probability.md
retrieved: 2026-10-19
---

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# Probability

37 topics in Mathematics, in the order they build on each other.

Page: https://lightmysky.com/learn/mathematics/areas/probability

- [Likelihood Language](https://lightmysky.com/learn/mathematics/likelihood-language-mt_iFFKZd-Vgv): Use probability language to describe and compare the likelihood of everyday events using words such as certain, likely, even chance, unlikely, impossible
- [Ordering Likelihoods](https://lightmysky.com/learn/mathematics/ordering-likelihoods-mt_UcGn2hjhYU): Compare the likelihood of different events and order them from least to most likely, including situations with unequal outcomes such as a bag with more of one colour than another or a spinner with sections of different sizes. Explain reasoning using informal language.
- [Equally Likely Outcomes](https://lightmysky.com/learn/mathematics/equally-likely-outcomes-mt_xSgAgg9Ej_): Understand that equally likely means every outcome has exactly the same chance; identify equal and unequal chances in fair coins, fair dice, equal-section spinners, unequal-section spinners, and bags with different colour counts.
- [Simple Chance Experiments](https://lightmysky.com/learn/mathematics/simple-chance-experiments-mt_J4j7d3iAfg): Conduct simple probability experiments — flipping a coin, rolling a die, pulling coloured counters from a bag — record results, and compare experimental outcomes with expected theoretical outcomes
- [Probability as a Fraction](https://lightmysky.com/learn/mathematics/probability-as-a-fraction-mt_-c4Ca_nBzX): Describe the probability of simple equally likely outcomes with fractions, including unit fractions such as 1/6, 1/2, and 1/3, and place these probabilities on a 0-to-1 scale.
- [The 0-to-1 Probability Scale](https://lightmysky.com/learn/mathematics/the-0-to-1-probability-scale-mt_t0g2SlP404): Understand probability as a measure expressed as a number between 0 (impossible) and 1 (certain); place events on the probability scale; express probabilities as fractions, decimals, and percentages
- [Calculating Simple Probability](https://lightmysky.com/learn/mathematics/calculating-simple-probability-mt_Zt30Gxi-qp): Calculate the probability of a simple event with equally likely outcomes using the formula: probability = number of favourable outcomes ÷ total number of possible outcomes; express the result as a fraction in its simplest form; apply to rolling dice, drawing from bags, and other simple chance situations
- [Experimental vs Theoretical](https://lightmysky.com/learn/mathematics/experimental-vs-theoretical-mt_-bMnJcPJy8): Compare observed relative frequencies with a probability model. With independent trials and fixed outcome probabilities matching the model, larger samples usually give closer estimates. A particular longer run can still be farther away.
- [Probabilities Sum to One](https://lightmysky.com/learn/mathematics/probabilities-sum-to-one-mt_3fwYu7imd4): Understand that when all possible outcomes of a trial are listed, their probabilities must add up to 1; use this to find the probability of an event NOT happening: P(not A) = 1 − P(A); apply this shortcut to avoid counting all unfavourable outcomes directly
- [Venn Diagrams and Counting Outcomes](https://lightmysky.com/learn/mathematics/venn-diagrams-and-counting-outcomes-mt_eSv_w46u6H): Construct and interpret Venn diagrams with two or three sets to organise and count outcomes; use systematic listing and the product rule for counting to enumerate all possible outcomes of combined events
- [Counting Outcomes with the Product Rule](https://lightmysky.com/learn/mathematics/counting-outcomes-with-the-product-rule-mt_74gSbYIG8p): Count the outcomes of several choices made in a row by multiplying the number of options at each stage, instead of listing them all.
- [The Addition Rule and Mutually Exclusive Events](https://lightmysky.com/learn/mathematics/the-addition-rule-and-mutually-exclusive-events-mt_QsIusFefeE): Use P(A or B) = P(A) + P(B) - P(A and B), and see why the subtraction disappears when two events cannot happen together.
- [Independent Events and the Multiplication Rule](https://lightmysky.com/learn/mathematics/independent-events-and-the-multiplication-rule-mt_MK8EPHEUdN): Use P(A and B) = P(A) × P(B) for independent events, and check whether learning one event happened changes the other's probability.
- [Discrete Random Variables and Probability Distributions](https://lightmysky.com/learn/mathematics/discrete-random-variables-and-probability-distributions-mt_yKztNkvvq3): A random variable attaches a number to each outcome, and its distribution lists every value the variable can take with the probability of taking it. Those probabilities have to total 1.
- [Probability Spaces: Sample Spaces, Events and Axioms](https://lightmysky.com/learn/mathematics/probability-spaces-sample-spaces-events-and-axioms-mt_4gsBeqJtIQ): Probability has a sample space, events as subsets, and a measure governed by three axioms: it is nonnegative, totals one and adds over disjoint events. Every rule met earlier follows from these axioms.
- [The Probability Scale](https://lightmysky.com/learn/mathematics/the-probability-scale-mt_vmQJAtAFuy): Understand probability as a measure on a scale from 0 (impossible) to 1 (certain); use the language of probability including likely, unlikely, certain, and impossible
- [Complementary events](https://lightmysky.com/learn/mathematics/complementary-events-mt_XfyqXLqzpx): Understand and apply the rule that probabilities of all mutually exclusive outcomes sum to one; use this to find the probability of a complementary event (P(not A) = 1 − P(A))
- [Experimental probability](https://lightmysky.com/learn/mathematics/experimental-probability-mt_sHJqh6UUya): Record, describe, and analyse the frequency of outcomes from probability experiments to develop an understanding of relative frequency as an estimate of probability
- [Sets & Venn Diagrams](https://lightmysky.com/learn/mathematics/sets-and-venn-diagrams-mt_1YwOCMMwD8): Enumerate sets and their unions and intersections systematically using tables, grids, and Venn diagrams to organise and count outcomes
- [Tree diagrams](https://lightmysky.com/learn/mathematics/tree-diagrams-mt_GDtFU5fyUv): Generate theoretical sample spaces for single and combined events using listing, tables, and tree diagrams, and calculate theoretical probabilities as the number of favourable outcomes divided by the total number of equally likely outcomes
- [Relative Frequency and Expected Outcomes](https://lightmysky.com/learn/mathematics/relative-frequency-and-expected-outcomes-mt_OQCnzkcHUH): Estimate a probability from experimental results, and use a probability to predict how many times an outcome should happen in a set number of trials.
- [Tree Diagrams Without Replacement](https://lightmysky.com/learn/mathematics/tree-diagrams-without-replacement-mt_NfiAY_wMwZ): Draw tree diagrams for two or three stages where the first outcome changes the second, multiply along a branch, and add the routes that fit the question.
- [Conditional Probability from Tables and Venn Diagrams](https://lightmysky.com/learn/mathematics/conditional-probability-from-tables-and-venn-diagrams-mt_Vx88ugViYP): Work out the probability of one event given that another has happened, by restricting attention to the row, column or region the condition allows.
- [Expected Value and Variance of a Discrete Random Variable](https://lightmysky.com/learn/mathematics/expected-value-and-variance-of-a-discrete-random-variable-mt_PFdo0Nm3FP): Weight each value by its probability to find the expected value. Calculate variance as E(X²) minus the square of E(X), and compare a game's stake with its expected payout to judge mathematical fairness.
- [The Binomial Distribution](https://lightmysky.com/learn/mathematics/the-binomial-distribution-mt_X2kVGK67Md): Count successes in a fixed number of independent trials that each have the same probability. The number of arrangements times the probability of one arrangement gives each term.
- [The Normal Distribution and Standardised Scores](https://lightmysky.com/learn/mathematics/the-normal-distribution-and-standardised-scores-mt_Cs32DT9KnA): A continuous symmetric model where most values sit near the mean. Converting a value to z = (x - μ) / σ says how many standard deviations from the mean it is, which makes different data sets comparable.
- [Finding Probabilities and Values from a Normal Model](https://lightmysky.com/learn/mathematics/finding-probabilities-and-values-from-a-normal-model-mt_naXMRySvos): Get the probability of a range of values from a normal model, and go the other way from a stated probability back to the cut-off value, sketching the region every time.
- [Conditional Probability, Independence and Bayes' Theorem](https://lightmysky.com/learn/mathematics/conditional-probability-independence-and-bayes-theorem-mt_wIi_XRzwwK): Conditioning restricts the sample space. Independence is the case where it changes nothing, and Bayes' theorem reverses the conditioning, which is how evidence updates a belief.
- [Discrete Models Beyond the Binomial](https://lightmysky.com/learn/mathematics/discrete-models-beyond-the-binomial-mt_9zc5xYjDg_): Geometric, hypergeometric and Poisson models make different assumptions. Choosing a model means deciding which assumptions fit the story before using any formula.
- [Continuous Random Variables and Density Functions](https://lightmysky.com/learn/mathematics/continuous-random-variables-and-density-functions-mt_jlf4zDTeNm): For a continuous variable, probability is area under a density, so any single value has probability zero. The cumulative function and the density are linked by calculus in both directions.
- [Expectation and Variance by Integration](https://lightmysky.com/learn/mathematics/expectation-and-variance-by-integration-mt_kRSaKgp3bs): The mean and variance of a continuous variable are integrals against the density, and the same linearity rules from the discrete case still hold.
- [Joint Distributions, Covariance and Independence](https://lightmysky.com/learn/mathematics/joint-distributions-covariance-and-independence-mt_YyTFTAqhSF): Two variables have a joint distribution, from which marginals follow. Covariance measures how they move together, and independence forces zero covariance without the converse holding.
- [Conditional Expectation Given a Sigma-Algebra](https://lightmysky.com/learn/mathematics/conditional-expectation-given-a-sigma-algebra-mt_i2sxZc-ER8): Define conditioning on information rather than on an event, as the best prediction measurable with respect to what is known.
- [Markov Chains and the Memoryless Assumption](https://lightmysky.com/learn/mathematics/markov-chains-and-the-memoryless-assumption-mt_2wDa_JxJ_o): The next state depends on the current state and on nothing before it. That single assumption turns a process into a matrix, and questions about the far future become questions about matrix powers.
- [Stationary Distributions and Long-Run Behaviour](https://lightmysky.com/learn/mathematics/stationary-distributions-and-long-run-behaviour-mt_MGfHvUGkeh): A distribution left unchanged by one step is an eigenvector of the transition matrix for eigenvalue one. Irreducibility and aperiodicity are the conditions under which every starting distribution converges to it.
- [Martingales and Optional Stopping](https://lightmysky.com/learn/mathematics/martingales-and-optional-stopping-mt_198uXX3jRJ): A martingale is a process whose expected next value, given everything so far, is its current value. Stopping it at a random time preserves that expectation, as long as the stopping rule is not allowed to look ahead.
- [Brownian Motion and Its Defining Properties](https://lightmysky.com/learn/mathematics/brownian-motion-and-its-defining-properties-mt_T5KtWIk07Z): Independent normal increments, continuous paths, and no derivative anywhere. It is the scaling limit of a random walk, and it is the process most continuous-time models are written against.