Visual Tools
Calculators
Tables
Mathematical Keyboard
Converters
Other Tools

Probability Terms and Definitions


About This Glossary

This glossary organizes 57 probability terms into eleven categories that together cover the vocabulary of probability theory from first principles through multivariate analysis.

Foundations establishes the starting point with 8 entries: probability itself, random experiments, sample spaces, events, elementary events, relative frequency, probability measures, and equally likely events.

Conditional Probability & Independence covers 3 entries on how events interact: conditional probability, independent events, and mutual exclusiveness.

Random Variables defines 5 entries bridging outcomes to numbers: Bernoulli experiments, sequences of Bernoulli trials, and the discrete/continuous random variable distinction.

Distribution Functions addresses 3 entries on how probability is described: cumulative distribution functions, probability mass functions, and probability density functions.

Measures spans 8 entries on numerical summaries: expected value, variance, standard deviation, covariance, correlation coefficient, conditional expectation, conditional variance, and moments.

Discrete Distributions covers 7 named distributions: Bernoulli, binomial, Poisson, discrete uniform, geometric, hypergeometric, and negative binomial.

Continuous Distributions presents 2 entries: exponential and normal distributions.

Multivariate Probability spans 11 entries on joint behavior: bivariate and n-variate random variables, independence, orthogonality, uncorrelatedness, marginal distributions, joint CDFs, PMFs, PDFs, and conditional PMFs and PDFs.

Transformations covers 3 entries: functions of random variables, PDFs of transformed variables, and moment generating functions.

Set Operations defines 6 entries: Venn diagrams, null sets, unions, intersections, disjoint sets, and complements.

Visual Tools includes the probability tree diagram.

Each definition includes intuitive explanations, key properties, and links to detailed lesson pages. Use the search bar or category filters above to navigate.
Conditional Probability & IndependenceContinuous DistributionsDiscrete DistributionsDistribution FunctionsFoundationsMeasuresMultivariate ProbabilityRandom VariablesSet OperationsTransformationsVisual Tools
BBernoulli DistributionDiscrete DistributionsBBernoulli ExperimentRandom VariablesBBinomial DistributionDiscrete DistributionsBBivariate Random VariableMultivariate ProbabilityCComplement of a SetSet OperationsCConditional ExpectationMeasuresCConditional ProbabilityConditional Probability & IndependenceCConditional Probability Density FunctionMultivariate ProbabilityCConditional Probability Mass FunctionMultivariate ProbabilityCConditional VarianceMeasuresCContinuous Random VariableRandom VariablesCCorrelation CoefficientMeasuresCCovarianceMeasuresCCumulative Distribution FunctionDistribution FunctionsDDiscrete Random VariableRandom VariablesDDiscrete Uniform DistributionDiscrete DistributionsDDisjoint SetsSet OperationsEElementary EventFoundationsEEqually Likely EventsFoundationsEEventFoundationsEExpected ValueMeasuresEExponential DistributionContinuous DistributionsFFunction of a Random VariableTransformationsGGeometric DistributionDiscrete DistributionsHHypergeometric DistributionDiscrete DistributionsIIndependent EventsConditional Probability & IndependenceIIndependent Random VariablesMultivariate ProbabilityIIntersection of SetsSet OperationsJJoint Cumulative Distribution FunctionMultivariate ProbabilityJJoint Probability Density FunctionMultivariate ProbabilityJJoint Probability Mass FunctionMultivariate ProbabilityMMarginal DistributionMultivariate ProbabilityMMoment Generating FunctionTransformationsMMoment of a Random VariableMeasuresMMutual ExclusivenessConditional Probability & IndependenceNN-Variate Random VariablesMultivariate ProbabilityNNegative Binomial DistributionDiscrete DistributionsNNormal DistributionContinuous DistributionsNNull SetSet OperationsOOrthogonal Random VariablesMultivariate ProbabilityPPDF of a Transformed VariableTransformationsPPoisson DistributionDiscrete DistributionsPProbabilityFoundationsPProbability Density FunctionDistribution FunctionsPProbability Mass FunctionDistribution FunctionsPProbability MeasureFoundationsPProbability TreeVisual ToolsRRandom ExperimentFoundationsRRandom VariableRandom VariablesRRelative FrequencyFoundationsSSample SpaceFoundationsSSequence of Bernoulli TrialsRandom VariablesSStandard DeviationMeasuresUUncorrelated Random VariablesMultivariate ProbabilityUUnion of SetsSet OperationsVVarianceMeasuresVVenn DiagramSet Operations
Conditional Probability & Independence(3)
Continuous Distributions(2)
Discrete Distributions(7)
Distribution Functions(3)
Foundations(8)
Measures(8)
Multivariate Probability(11)
Random Variables(5)
Set Operations(6)
Transformations(3)
Visual Tools(1)
57 of 57 terms
Conditional Probability & Independence
Conditional ProbabilityIndependent EventsMutual Exclusiveness
Continuous Distributions
Exponential DistributionNormal Distribution
Discrete Distributions
Negative Binomial DistributionBernoulli DistributionBinomial DistributionPoisson DistributionDiscrete Uniform DistributionGeometric DistributionHypergeometric Distribution
Distribution Functions
Cumulative Distribution FunctionProbability Mass FunctionProbability Density Function
Foundations
ProbabilityRandom ExperimentSample SpaceEventElementary EventRelative FrequencyProbability MeasureEqually Likely Events
Measures
Expected ValueVarianceStandard DeviationCovarianceCorrelation CoefficientConditional ExpectationConditional VarianceMoment of a Random Variable
Multivariate Probability
Bivariate Random VariableN-Variate Random VariablesIndependent Random VariablesOrthogonal Random VariablesUncorrelated Random VariablesMarginal DistributionJoint Cumulative Distribution FunctionJoint Probability Mass FunctionJoint Probability Density FunctionConditional Probability Mass FunctionConditional Probability Density Function
Random Variables
Bernoulli ExperimentSequence of Bernoulli TrialsRandom VariableDiscrete Random VariableContinuous Random Variable
Set Operations
Venn DiagramNull SetUnion of SetsIntersection of SetsDisjoint SetsComplement of a Set
Transformations
Function of a Random VariablePDF of a Transformed VariableMoment Generating Function
Visual Tools
Probability Tree

57 terms

Discrete Distributions

(7 items)

Negative Binomial Distribution

A discrete distribution counting the number of trials needed to achieve a fixed number rr of successes in a sequence of independent Bernoulli trials with success probability pp.
See details
↑ Back to top
intuitionnotationrelated concepts
The negative binomial distribution answers "how many trials until rr successes?" It generalizes the geometric distribution, which is the special case r=1r = 1.
↑ Back to top

Bernoulli Distribution

A 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.
See details
↑ Back to top
intuitionpropertiesrelated concepts
The Bernoulli distribution models a single yes/no trial. It is the simplest discrete distribution and the building block for the binomial, geometric, and negative binomial distributions.
↑ Back to top

Binomial Distribution

A discrete distribution counting the number of successes in nn independent Bernoulli trials, each with success probability pp.
See details
↑ Back to top
intuitionnotationrelated concepts
The binomial distribution answers "how many successes?" in a fixed number of independent trials. It requires a fixed number of trials, constant probability, and independence.
↑ Back to top

Poisson Distribution

A discrete distribution modelling the number of events occurring in a fixed interval, where events happen independently at a constant average rate λ\lambda.
See details
↑ Back to top
intuitionnotationrelated concepts
The Poisson distribution counts rare events in a fixed window of time or space. It applies when events occur independently and the average rate is known.
↑ Back to top

Discrete Uniform Distribution

A discrete distribution where each of nn possible values has equal probability 1/n1/n.
See details
↑ Back to top
intuitionnotationrelated concepts
The discrete uniform distribution applies when every outcome is equally likely — the probability counterpart of "fair." A fair die is the classic example.
↑ Back to top

Geometric Distribution

A discrete distribution counting the number of trials needed to obtain the first success in a sequence of independent Bernoulli trials with success probability pp.
See details
↑ Back to top
intuitionnotationrelated concepts
The geometric distribution answers "how long until the first success?" It is the discrete analogue of the exponential distribution and shares its memoryless property.
↑ Back to top

Hypergeometric Distribution

A discrete distribution describing the number of successes in nn draws without replacement from a finite population containing KK successes and N−KN - K failures.
See details
↑ Back to top
intuitionnotationrelated concepts
The hypergeometric distribution is used when sampling without replacement. Unlike the binomial, the probability of success changes from draw to draw because the population is finite.
↑ Back to top

Foundations

(8 items)

Probability

A 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.
See details
↑ Back to top
intuitionnotationrelated concepts
Probability quantifies how likely an event is to occur. A value of 0 means impossible, 1 means certain, and values in between reflect degrees of likelihood.
↑ Back to top

Random Experiment

A process or action whose outcome cannot be predicted with certainty before it is performed.
See details
↑ Back to top
intuitionexamplesrelated concepts
A random experiment is any procedure that produces an unpredictable result. Rolling a die, drawing a card, or measuring a physical quantity are all random experiments.
↑ Back to top

Sample Space

Ω={ω1,ω2,…}\Omega = \{\omega_1, \omega_2, \ldots\} — the set of all possible outcomes of a random experiment.
See details
↑ Back to top
intuitionnotationrelated concepts
The sample space is the complete list of everything that can happen. Every probability question begins by identifying this set, because events and probabilities are defined relative to it.
↑ Back to top

Event

A⊆ΩA \subseteq \Omega — a subset of the sample space.
See details
↑ Back to top
intuitionexamplesrelated concepts
An event is a collection of outcomes we are interested in. It can contain one outcome, several outcomes, or even all outcomes. Probabilities are assigned to events, not to individual outcomes directly.
↑ Back to top

Elementary Event

An event consisting of exactly one outcome: {ω}\{\omega\} where ω∈Ω\omega \in \Omega.
See details
↑ Back to top
intuitionrelated concepts
An elementary event is the simplest possible event — a single outcome from the sample space that cannot be broken down further.
↑ Back to top

Relative Frequency

fn(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.
See details
↑ Back to top
intuitionrelated concepts
Relative frequency is the proportion of times an event occurs in repeated experiments. As the number of trials grows, relative frequency tends to stabilize near the true probability of the event.
↑ Back to top

Probability Measure

A 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 additivity for disjoint events.
See details
↑ Back to top
intuitionpropertiesrelated concepts
A probability measure is the formal rule that assigns a number between 0 and 1 to every event in a way that is internally consistent. It is the mathematical object that makes probability rigorous.
↑ Back to top

Equally Likely Events

Events 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).
See details
↑ Back to top
intuitionexamplesrelated concepts
When all outcomes in a finite sample space have the same probability, they are equally likely. In this case probability reduces to counting: P(A)=∣A∣/∣Ω∣P(A) = |A| / |\Omega|.
↑ Back to top

Conditional Probability & Independence

(3 items)

Conditional Probability

P(A∣B)=P(A∩B)P(B)P(A \mid B) = \frac{P(A \cap B)}{P(B)}, defined when P(B)>0P(B) > 0.
See details
↑ Back to top
intuitionnotationrelated concepts
Conditional probability measures the likelihood of an event given that another event has already occurred. It restricts attention to a smaller part of the sample space.
↑ Back to top

Independent Events

Events AA and BB are independent if and only if P(A∩B)=P(A)⋅P(B)P(A \cap B) = P(A) \cdot P(B).
See details
↑ Back to top
intuitioncommon errorsrelated concepts
Two events are independent when the occurrence of one provides no information about the other. Knowing that BB happened does not change the probability of AA.
↑ Back to top

Mutual Exclusiveness

Events AA and BB are mutually exclusive if A∩B=∅A \cap B = \emptyset.
See details
↑ Back to top
intuitionpropertiesrelated concepts
Mutually exclusive events cannot happen at the same time. If one occurs, the other is automatically ruled out.
↑ Back to top

Random Variables

(5 items)

Bernoulli Experiment

A random experiment with exactly two possible outcomes, conventionally called success (SS) and failure (FF).
See details
↑ Back to top
intuitionexamplesrelated concepts
A Bernoulli experiment is the simplest random experiment: something either happens or it does not. It is the building block for more complex models like the binomial distribution.
↑ Back to top

Sequence of Bernoulli Trials

A sequence of independent Bernoulli experiments, each with the same success probability pp.
See details
↑ Back to top
intuitionrelated concepts
Repeating the same yes/no experiment independently under identical conditions. The number of successes, the trial of first success, and similar quantities each give rise to a named distribution.
↑ Back to top

Random Variable

X:Ω→RX: \Omega \to \mathbb{R} — a function that assigns a real number to each outcome in the sample space.
See details
↑ Back to top
intuitionnotationrelated concepts
A random variable translates outcomes of a random experiment into numbers. This numerical representation makes it possible to compute averages, measure spread, and define distributions.
↑ Back to top

Discrete Random Variable

A random variable whose set of possible values is finite or countably infinite.
See details
↑ Back to top
intuitionexamplesrelated concepts
A discrete random variable takes on isolated, separated values that can be listed — even if the list is infinite. Its probability distribution is described by a probability mass function.
↑ Back to top

Continuous Random Variable

A random variable whose set of possible values forms an interval or union of intervals on the real line.
See details
↑ Back to top
intuitionexamplesrelated concepts
A continuous random variable can take any value within a range. Probability is spread smoothly rather than concentrated at individual points, and is described by a probability density function.
↑ Back to top

Distribution Functions

(3 items)

Cumulative Distribution Function

FX(x)=P(X≤x)F_X(x) = P(X \le x) for all x∈Rx \in \mathbb{R}.
See details
↑ Back to top
intuitionpropertiesrelated concepts
The CDF tracks how much probability has accumulated up to each value. It answers "how likely is the random variable to be at most xx?" and works for any type of distribution.
↑ Back to top

Probability Mass Function

pX(x)=P(X=x)p_X(x) = P(X = x) — the probability that a discrete random variable XX takes the value xx.
See details
↑ Back to top
intuitionpropertiesrelated concepts
The PMF assigns a probability to each individual value a discrete random variable can take. All values are non-negative and their sum equals 1.
↑ Back to top

Probability Density Function

A function fX(x)≥0f_X(x) \ge 0 such that P(a≤X≤b)=∫abfX(x) dxP(a \le X \le b) = \int_a^b f_X(x)\,dx and ∫−∞∞fX(x) dx=1\int_{-\infty}^{\infty} f_X(x)\,dx = 1.
See details
↑ Back to top
intuitioncommon errorsrelated concepts
The PDF describes how densely probability is spread across values of a continuous random variable. Its value at a point is not a probability — probabilities come from integrating the PDF over intervals.
↑ Back to top

Measures

(8 items)

Expected Value

E[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).
See details
↑ Back to top
intuitionpropertiesrelated concepts
The expected value is the long-run average of a random variable over many repetitions. It represents the center of mass of the distribution.
↑ Back to top

Variance

Var⁡(X)=E[(X−μ)2]\operatorname{Var}(X) = E[(X - \mu)^2] where μ=E[X]\mu = E[X].
See details
↑ Back to top
intuitionpropertiesrelated concepts
Variance measures how spread out a random variable's values are around its expected value. A larger variance means outcomes are more dispersed.
↑ Back to top

Standard Deviation

σX=Var⁡(X)\sigma_X = \sqrt{\operatorname{Var}(X)}
See details
↑ Back to top
intuitionrelated concepts
Standard deviation is the square root of variance. It measures spread in the same units as the random variable, making it easier to interpret than variance.
↑ Back to top

Covariance

Cov⁡(X,Y)=E[(X−E[X])(Y−E[Y])]\operatorname{Cov}(X, Y) = E[(X - E[X])(Y - E[Y])]
See details
↑ Back to top
intuitionpropertiesrelated concepts
Covariance measures how two random variables move together. Positive covariance means they tend to increase together; negative means one tends to increase when the other decreases.
↑ Back to top

Correlation Coefficient

ρXY=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.
See details
↑ Back to top
intuitionpropertiesrelated concepts
The correlation coefficient normalizes covariance to a scale between −1-1 and 11. It measures the strength and direction of the linear relationship between two random variables.
↑ Back to top

Conditional Expectation

E[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).
See details
↑ Back to top
intuitionrelated concepts
Conditional expectation is the expected value of one random variable given that another takes a specific value. It adjusts the average to account for known information.
↑ Back to top

Conditional Variance

Var⁡(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]
See details
↑ Back to top
intuitionrelated concepts
Conditional variance measures the spread of one random variable around its conditional mean, given a specific value of another variable.
↑ Back to top

Moment of a Random Variable

The 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].
See details
↑ Back to top
intuitionrelated concepts
Moments are numerical summaries of a distribution's shape. The first moment is the mean, the second central moment is the variance, and higher moments capture skewness and tail behavior.
↑ Back to top

Continuous Distributions

(2 items)

Exponential Distribution

A continuous distribution describing the time between events in a process where events occur independently at a constant rate λ\lambda.
See details
↑ Back to top
intuitionnotationrelated concepts
The exponential distribution models waiting time — how long until the next event. Its defining feature is the memoryless property: the probability of waiting longer does not depend on how long you have already waited.
↑ Back to top

Normal Distribution

A continuous distribution characterized by a symmetric, bell-shaped curve, fully determined by its mean μ\mu and variance σ2\sigma^2.
See details
↑ Back to top
intuitionnotationrelated concepts
The normal distribution appears when many small independent effects combine. Its bell curve is symmetric about the mean, with probability decreasing smoothly in both directions.
↑ Back to top

Multivariate Probability

(11 items)

Bivariate Random Variable

A pair of random variables (X,Y)(X, Y) defined on the same sample space, considered jointly.
See details
↑ Back to top
intuitionrelated concepts
A bivariate random variable treats two measurements taken from the same experiment as a single object. Their joint behavior — how they relate, co-occur, or depend on each other — is captured by a joint distribution.
↑ Back to top

N-Variate Random Variables

A vector (X1,X2,…,Xn)(X_1, X_2, \ldots, X_n) of nn random variables defined on the same sample space.
See details
↑ Back to top
intuitionrelated concepts
An extension of bivariate random variables to any number of components. The joint distribution describes how all nn variables behave together.
↑ Back to top

Independent Random Variables

Random 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.
See details
↑ Back to top
intuitionpropertiesrelated concepts
Independent random variables carry no information about each other. Knowing the value of one does not change the distribution of the other.
↑ Back to top

Orthogonal Random Variables

Random variables XX and YY are orthogonal if E[XY]=0E[XY] = 0.
See details
↑ Back to top
intuitionrelated concepts
Orthogonality is an algebraic condition on the product of two random variables. It is weaker than independence and does not imply zero covariance unless the means are zero.
↑ Back to top

Uncorrelated Random Variables

Random 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].
See details
↑ Back to top
intuitioncommon errorsrelated concepts
Uncorrelated means no linear relationship between two variables. Independent random variables are always uncorrelated, but uncorrelated variables can still be dependent through nonlinear relationships.
↑ Back to top

Marginal Distribution

The distribution of one random variable obtained from a joint distribution by summing (discrete) or integrating (continuous) over all values of the other variable(s).
See details
↑ Back to top
intuitionrelated concepts
A marginal distribution extracts the standalone behavior of one variable from a joint distribution, ignoring the other variables. In a contingency table, marginals appear as row or column totals.
↑ Back to top

Joint Cumulative Distribution Function

FX,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}.
See details
↑ Back to top
intuitionrelated concepts
The joint CDF extends the cumulative distribution function to multiple random variables. It gives the probability that all variables simultaneously fall below their respective thresholds.
↑ Back to top

Joint Probability Mass Function

pX,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.
See details
↑ Back to top
intuitionpropertiesrelated concepts
The joint PMF assigns a probability to each specific combination of values two discrete random variables can take simultaneously.
↑ Back to top

Joint Probability Density Function

A 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.
See details
↑ Back to top
intuitionrelated concepts
The joint PDF describes how probability density is spread over the plane for two continuous random variables. Probabilities are obtained by integrating over regions, not by evaluating at points.
↑ Back to top

Conditional Probability Mass Function

pX∣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.
See details
↑ Back to top
intuitionrelated concepts
The conditional PMF gives the probability distribution of one discrete random variable when another discrete random variable is known to take a specific value.
↑ Back to top

Conditional Probability Density Function

fX∣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.
See details
↑ Back to top
intuitionrelated concepts
The conditional PDF describes the density of one continuous random variable when another continuous random variable is known to take a specific value.
↑ Back to top

Transformations

(3 items)

Function of a Random Variable

Given a random variable XX and a function gg, Y=g(X)Y = g(X) defines a new random variable whose distribution is determined by the distribution of XX and the function gg.
See details
↑ Back to top
intuitionrelated concepts
Applying a function to a random variable produces a new random variable. The challenge is determining the distribution of the result from the distribution of the original.
↑ Back to top

PDF of a Transformed Variable

If Y=g(X)Y = g(X) where gg is monotone and differentiable, then fY(y)=fX(g−1(y))⋅∣ddyg−1(y)∣f_Y(y) = f_X(g^{-1}(y)) \cdot |\frac{d}{dy} g^{-1}(y)|.
See details
↑ Back to top
intuitionrelated concepts
When a continuous random variable is transformed, its density changes. The change-of-variables formula accounts for both the mapping of values and the stretching or compression of the density.
↑ Back to top

Moment Generating Function

MX(t)=E[etX]M_X(t) = E[e^{tX}], defined for real values of tt where the expectation exists.
See details
↑ Back to top
intuitionpropertiesrelated concepts
The moment generating function encodes all moments of a random variable in a single function. The kk-th moment is obtained by differentiating kk times and evaluating at t=0t = 0.
↑ Back to top

Set Operations

(6 items)

Venn Diagram

A graphical representation using overlapping circles to depict sets (events) and their relationships within a sample space.
See details
↑ Back to top
intuitionrelated concepts
Venn diagrams visualize how events overlap, combine, or exclude each other. They make set operations like union, intersection, and complement immediately visible.
↑ Back to top

Null Set

∅\emptyset — the set containing no elements, representing an impossible event in probability.
See details
↑ Back to top
intuitionrelated concepts
The null set is the event that can never occur. Its probability is always zero: P(∅)=0P(\emptyset) = 0.
↑ Back to top

Union of Sets

A∪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.
See details
↑ Back to top
intuitionrelated concepts
The union combines two events into one that occurs whenever either event (or both) occurs.
↑ Back to top

Intersection of Sets

A∩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.
See details
↑ Back to top
intuitionrelated concepts
The intersection is the event where both conditions are met at the same time.
↑ Back to top

Disjoint Sets

Sets AA and BB are disjoint if A∩B=∅A \cap B = \emptyset — they share no common elements.
See details
↑ Back to top
intuitionrelated concepts
Disjoint sets have no overlap. In probability, disjoint events are mutually exclusive — they cannot both occur in the same trial.
↑ Back to top

Complement of a Set

Ac={ω∈Ω:ω∉A}A^c = \{\omega \in \Omega : \omega \notin A\} — all outcomes in the sample space that are not in AA.
See details
↑ Back to top
intuitionrelated concepts
The complement of an event is everything that can happen except that event. P(Ac)=1−P(A)P(A^c) = 1 - P(A).
↑ Back to top

Visual Tools

(1 items)

Probability Tree

A branching diagram where each node represents a stage of a sequential random process, branches represent possible outcomes, and branch labels show conditional probabilities.
See details
↑ Back to top
intuitionrelated concepts
A probability tree lays out a multi-stage experiment visually. Multiplying along a path gives joint probabilities; summing across paths reaching the same outcome gives marginal probabilities.
↑ Back to top