Probability Diagrams

The 36 outcomes of two dice, each with probability 1/36

Probability Axioms: Foundation of Probability Theory › The Probability Axioms (Overview)

Branches below A carry P(B given A) and P(not B given A)

Bayes' Theorem › Connection to Conditional Probability

One outcome highlighted across every branch of the partition

Bayes' Theorem › The Role of Total Probability

Binomial CDF, n = 10, p = 0.5: a staircase

CDF (Cumulative Distribution Function) › Discrete Random Variables

Normal CDF, mean 0, standard deviation 1

CDF (Cumulative Distribution Function) › Continuous Random Variables

Continuous uniform CDF, a = 0, b = 10: a straight ramp

CDF (Cumulative Distribution Function) › Using the CDF to Compute Probabilities

The standard normal density, the limit shape of averages

Central Limit Theorem: Sample Means and Normal Distribution › Central Limit Theorem Notation

Sum equals 7: six of the 36 equally likely cells

Combinatorics in Probability: Counting and Classical Formula › Finite Sample Spaces and Counting

Three tosses: the eight sequences of heads and tails

Combinatorics in Probability: Counting and Classical Formula › Counting in Classical Probability Experiments

Given A: only the subtree below A remains

Conditional Probability: P(A|B) Formula and Examples › Formal Meaning of Conditional Probability

Event A across three compartments, the middle one selected

Conditional Probability: P(A|B) Formula and Examples › Visual Representations

A 2 by 2 contingency table with row and column totals

Conditional Probability: P(A|B) Formula and Examples › Examples

Even sum: 18 of the 36 outcomes of two dice

Events in Probability: Sample Space and Event Operations › Sample Space and Events

Two events A and B with their intersection selected

Events in Probability: Sample Space and Event Operations › Set-Theoretic View

Given A: the subtree below A

Events in Probability: Sample Space and Event Operations › Conditional Events

A distribution pulled to the right: the mean follows the weight

Expected Value Page › The Intuition Behind Expected Value

A discrete distribution with E[X] = 3.4

Expected Value Page › Expected Value for Discrete Random Variables (PMF)

A waffle chart with equal columns: independence

Independence of Events › Visual Representations

A 2 by 2 contingency table

Independence of Events › Common Mistakes

Majority heads: the four sequences where the indicator equals 1

Indicator Random Variables: Converting Events to 0-1 Variables › From Events to Random Variables

A 2 by 3 contingency table with marginal totals

Joint Probability: Combined Outcomes for Multiple Variables › Joint Probability Notation

One path through the tree: the joint probability of A and B

Joint Probability: Combined Outcomes for Multiple Variables › Connection to Conditional Probability

Standard normal density: the shape the central limit theorem adds

Law of Large Numbers: Sample Mean Convergence › LLN vs Central Limit Theorem

Exponential CDF, lambda = 1: the median is where F crosses one half

Median: 50th Percentile of Distributions › Median for Continuous Distributions

Normal CDF, mean 0, standard deviation 1

Median: 50th Percentile of Distributions › How to Find the Median

Poisson PMF, lambda = 3: two bars tie for the peak

Mode: Peak Probability Concentration › Mode for Discrete Distributions

Exponential density, lambda = 1: the peak sits at zero

Mode: Peak Probability Concentration › Mode for Continuous Distributions

Three tosses labelled by the number of heads

Random Variables: Mapping Outcomes to Numbers › From Outcomes to Numerical Values

Normal density: a continuous random variable

Random Variables: Mapping Outcomes to Numbers › Types of Random Variables

Binomial CDF, n = 10, p = 0.5: the statement X ≤ k

Random Variables: Mapping Outcomes to Numbers › Random Variables and Probability Statements

The sample space of two dice: 36 ordered outcomes

Sample Space: All Possible Outcomes in Probability › Definition of Sample Space

Even sum: an event as a subset of the sample space

Sample Space: All Possible Outcomes in Probability › Relationship to Events

Three coin tosses: eight equally likely sequences

Sample Space: All Possible Outcomes in Probability › Sample Space in Practice

Two events with their intersection selected

Sets in Probability: Foundation for Events and Models › Core Set Operations in Probability

Three events inside a sample space, the outside region selected

Sets in Probability: Foundation for Events and Models › How Sets Become Events

A three-way partition and its outcomes

Law of Total Probability: Weighted Case Analysis › Splitting a Probability Across Cases

Event A drawn across a three-compartment partition

Law of Total Probability: Weighted Case Analysis › Diagrammatic Representations

The first case highlighted: its contribution to the total

Law of Total Probability: Weighted Case Analysis › Why the Law Works

The subtree below A: branch weights are conditional probabilities

Tree Diagrams: Visualizing Sequential Probability › Tree Diagrams and Conditional Probability

One outcome highlighted across every branch

Tree Diagrams: Visualizing Sequential Probability › Tree Diagrams and the Law of Total Probability

One complete path: multiply along the branches

Tree Diagrams: Visualizing Sequential Probability › Using Tree Diagrams to Compute Probabilities

A dataset around its mean of 20

Variance: Measuring Spread in Probability › Variance Notation

The same dataset with the sample variance

Variance: Measuring Spread in Probability › Calculating Variance: General Case

A high-variance dataset

Variance: Measuring Spread in Probability › Variance vs Standard Deviation

Binomial PMF, n = 10, p = 0.5

Probability Distributions: Discrete & Continuous › Probability Function

Normal CDF, mean 0, standard deviation 1

Probability Distributions: Discrete & Continuous › Cumulative Distribution Function (CDF)

Continuous uniform density, a = 0, b = 10

Title › Continuous vs Discrete Distributions

Standard normal density

Title › Types of Continuous Distributions

Exponential CDF, lambda = 1

Title › Working with PDFs and CDFs

Exponential density, lambda = 1

Exponential Distribution: PDF, CDF & Properties › Probability Density Function (PDF) and Support (Range)

Exponential CDF, lambda = 1

Exponential Distribution: PDF, CDF & Properties › Cumulative Distribution Function (CDF)

Normal density, mean 0, standard deviation 1

Normal Distribution: Bell Curve, PDF & Z-Scores › Probability Density Function (PDF) and Support (Range)

Normal CDF, mean 0, standard deviation 1

Normal Distribution: Bell Curve, PDF & Z-Scores › Cumulative Distribution Function (CDF)

Continuous uniform density, a = 0, b = 10

Continuous Uniform Distribution: PDF, Mean & Variance › Probability Density Function (PDF) and Support (Range)

Continuous uniform CDF, a = 0, b = 10

Continuous Uniform Distribution: PDF, Mean & Variance › Cumulative Distribution Function (CDF)

Discrete uniform PMF, values 1 to 6

Title › What Makes a Distribution Discrete

Binomial CDF, n = 10, p = 0.5

Title › Discrete vs Continuous Distributions

Binomial PMF, n = 10, p = 0.5

Binomial Distribution: PMF, Mean & Variance › Probability Mass Function (PMF) and Support (Range)

Binomial CDF, n = 10, p = 0.5

Binomial Distribution: PMF, Mean & Variance › Cumulative Distribution Function (CDF)

Three tosses: eight sequences grouped by number of heads

Binomial Distribution: PMF, Mean & Variance › Applications and Examples

Geometric PMF, p = 0.3

Geometric Distribution: PMF, Mean & Memoryless › Probability Mass Function (PMF) and Support (Range)

Geometric CDF, p = 0.3

Geometric Distribution: PMF, Mean & Memoryless › Cumulative Distribution Function (CDF)

Hypergeometric PMF, N = 50, K = 20, n = 10

Hypergeometric Distribution: Sampling Without Replacement › Probability Mass Function (PMF) and Support (Range)

Hypergeometric CDF, N = 50, K = 20, n = 10

Hypergeometric Distribution: Sampling Without Replacement › Cumulative Distribution Function (CDF)

Negative binomial PMF, r = 3, p = 0.4

Negative Binomial Distribution: Trials Until r Successes › Probability Mass Function (PMF) and Support (Range)

Negative binomial PMF with the mean marked

Negative Binomial Distribution: Trials Until r Successes › Expected Value (Mean)

Poisson PMF, lambda = 3

Poisson Distribution: Event Counts & Rate Parameter › Probability Mass Function (PMF) and Support (Range)

Poisson CDF, lambda = 3

Poisson Distribution: Event Counts & Rate Parameter › Cumulative Distribution Function (CDF)

Discrete uniform PMF, a = 1 to b = 6

Discrete Uniform Distribution: Equal Probability Outcomes › Probability Mass Function (PMF) and Support (Range)

Discrete uniform CDF, a = 1 to b = 6

Discrete Uniform Distribution: Equal Probability Outcomes › Cumulative Distribution Function (CDF)

A dataset with its mean and squared deviations

Probability Inequalities: Bounds Without Full Distributions › What Inequalities Depend On

Chebyshev bound for a normal variable, mean 10, variance 4, deviation 3

Probability Inequalities: Bounds Without Full Distributions › Featured Inequalities

Chebyshev bound for a normal variable, mean 10, variance 4, deviation 3

Chebyshev's Inequality › Statement of Chebyshev's Inequality

Chebyshev bound for a uniform variable

Chebyshev's Inequality › Limitations of Chebyshev's Inequality

Markov bound for an exponential variable, mean 10, threshold 15

Markov Inequality › Statement of Markov's Inequality

A threshold below the mean: the bound exceeds 1 and says nothing

Markov Inequality › Limitations of Markov's Inequality

A probability model: three tosses, eight outcomes, each 1/8

Probability Models: Mathematical Frameworks for Randomness › What Is a Probability Model

Discrete uniform PMF, values 1 to 6

Probability Models: Mathematical Frameworks for Randomness › Models, Random Variables, and Distributions

Two dice: a finite model with 36 outcomes

Probability Models: Mathematical Frameworks for Randomness › Simple Discrete Models

Three tosses: the eight outcomes

Coin Toss Probability Model: Binary Outcomes and Distributions › Outcome Space

Majority heads: four of the eight sequences

Coin Toss Probability Model: Binary Outcomes and Distributions › Events

Binomial PMF, n = 10, p = 0.5: heads in ten tosses

Coin Toss Probability Model: Binary Outcomes and Distributions › Distributions Built from Repeated Coin Tosses

Two dice: 36 ordered outcomes

Dice Roll Probability Model: Finite Multi-Outcome Randomness › Outcome Space

Sum equals 7: six outcomes on the diagonal

Dice Roll Probability Model: Finite Multi-Outcome Randomness › Events

Discrete uniform PMF, a = 1 to b = 6

Dice Roll Probability Model: Finite Multi-Outcome Randomness › Distributions Directly Induced by a Single Roll