---
title: Probability & Statistics
description: Chance made rigorous: probability theory, hypothesis testing, Bayesian methods, and statistical inference.
category: mathematics
url: /subcategory/mathematics/probability-theory
---

# Probability & Statistics

Chance made rigorous: probability theory, hypothesis testing, Bayesian methods, and statistical inference.

## Available Subjects

### ANOVA

Analysis of Variance (ANOVA) is a statistical method used to compare means across three or more groups. Learners will understand how to partition variance, perform one-way and two-way ANOVA tests, and interpret post-hoc analyses.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 2 books, 1 courses, 2 websites, 1 papers

[Learn ANOVA](/subject/statistics-and-probability-anova)

### Actuarial Science

Actuaries price risk for insurers and pensions. You will learn the probability and financial mathematics behind the actuarial exams, life and non-life models, and the exam path itself.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 3 books, 2 websites

[Learn Actuarial Science](/subject/actuarial-science)

### Bayesian Statistics

Bayesian statistics is a mathematical framework that applies probability to statistical problems, updating beliefs as new data is observed. You will understand how to construct prior distributions, compute posterior probabilities, and perform Bayesian inference.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 1 videos, 3 books, 2 courses, 4 websites

[Learn Bayesian Statistics](/subject/bayesian-statistics)

### Bayes’ Theorem

Bayes' theorem is a mathematical formula for determining conditional probability based on prior knowledge of conditions related to an event. Learners will understand how to update probabilities as new evidence becomes available.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 2 websites, 1 papers

[Learn Bayes’ Theorem](/subject/bayes-theorem)

### Blackjack & Card Counting

Card counting is the advantage-play technique that turns blackjack's house edge into a player edge by tracking the ratio of high to low cards left in the shoe. You will learn basic strategy, the Hi-Lo count, bet spreading and bankroll management, and the risks of casino countermeasures.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 5 books, 3 websites

[Learn Blackjack & Card Counting](/subject/blackjack-card-counting)

### Conditional Probability

Conditional probability measures the likelihood of an event occurring given that another event has already occurred. Learners will understand how to calculate dependent probabilities and apply these concepts to real-world scenarios.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 3 websites

[Learn Conditional Probability](/subject/conditional-probability)

### Confidence Intervals

Confidence intervals provide an estimated range of values likely to contain an unknown population parameter. Learners will understand how to calculate and interpret these intervals to quantify the uncertainty and precision of statistical estimates.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 2 books, 1 courses, 1 websites, 2 papers

[Learn Confidence Intervals](/subject/statistics-and-probability-confidence-intervals)

### Descriptive Statistics

This field involves summarizing and organizing features of a dataset through numerical calculations and graphs. Learners will understand how to calculate measures of central tendency, dispersion, and distribution shape to describe data accurately.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 1 videos, 3 books, 1 courses

[Learn Descriptive Statistics](/subject/descriptive-statistics)

### Distributions

Distributions, or generalized functions, extend the concept of derivative to all continuous functions. Learners will understand how to formulate and solve weak solutions for partial differential equations that lack classical solutions.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 1 books, 2 courses, 2 websites

[Learn Distributions](/subject/distributions)

### Expected Value

Expected value is the long-run average value of a random variable over many repetitions of an experiment. Learners will understand how to calculate and interpret expected outcomes for discrete and continuous distributions.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 1 videos, 1 books, 2 websites

[Learn Expected Value](/subject/expected-value)

### Experimental Design

This discipline covers the planning and structuring of scientific experiments to ensure valid statistical analysis. Learners will understand how to define variables, control confounding factors, and select appropriate sampling methods to establish causal relationships.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 4 books, 1 courses, 2 websites

[Learn Experimental Design](/subject/experimental-design)

### Forecasting & Calibrated Thinking

Good forecasters think in probabilities and update on evidence. You will learn calibration, base rates, Bayesian updating, the superforecasting research and how to practise on prediction platforms.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 1 books, 3 websites, 1 papers

[Learn Forecasting & Calibrated Thinking](/subject/forecasting-and-rationality)

### Hypothesis Testing

Hypothesis testing is a framework for determining if experimental results support a specific theory. Learners will understand how to set up null and alternative hypotheses, select appropriate statistical tests, and interpret p-values and confidence levels.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 1 videos, 2 books, 1 websites

[Learn Hypothesis Testing](/subject/statistics-and-probability-hypothesis-testing)

### Limit Theorems

Limit theorems describe the behavior of sample statistics as the sample size grows infinitely large. Learners will understand the Law of Large Numbers and the Central Limit Theorem to make statistical inferences.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 1 books, 1 courses, 1 websites

[Learn Limit Theorems](/subject/limit-theorems)

### Markov Chains

Markov chains are mathematical models describing sequences of events where the probability of each event depends only on the state attained in the previous event. Learners will understand transition matrices, steady-state probabilities, and how to model stochastic systems.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 2 books

[Learn Markov Chains](/subject/markov-chains)

### Poker

Poker is a game of incomplete information, probability and psychology. You will learn hand ranges, pot odds, game theory optimal play, bankroll management and how to study the game seriously.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 6 books, 1 courses, 7 websites

[Learn Poker](/subject/poker)

### Probability

Probability is the mathematical study of uncertainty and randomness. Learners will understand random variables, probability distributions, Bayes' theorem, and how to calculate the likelihood of events to inform decision-making and statistical models.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 2 videos, 3 books, 3 courses, 2 websites

[Learn Probability](/subject/probability)

### Probability Distributions

This area of mathematics describes the likelihood of different outcomes in a random event. Learners will understand discrete and continuous distributions, including normal, binomial, and Poisson distributions, and how to apply them to model real-world scenarios.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 2 books, 3 websites

[Learn Probability Distributions](/subject/probability-distributions)

### Probability Spaces

A probability space is a mathematical construct that models a random process, consisting of a sample space, event space, and probability measure. Learners will understand how to rigorously define and analyze uncertainty using axiomatic probability theory.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 3 books, 1 websites

[Learn Probability Spaces](/subject/probability-spaces)

### Probability Theory

Probability theory is the mathematical framework for analyzing random phenomena and uncertainty. Learners will understand fundamental concepts like sample spaces, conditional probability, independent events, and random variables to solve complex predictive and analytical problems.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 3 books

[Learn Probability Theory](/subject/statistics-and-probability-probability-theory)

### Random Variables

Random variables are rules that associate numerical values with the outcomes of random processes. Learners will understand the distinction between discrete and continuous variables, compute expected values, and use cumulative distribution functions to analyze probabilistic events.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 1 books, 1 courses, 2 websites

[Learn Random Variables](/subject/random-variables)

### Regression Analysis

This statistical method estimates the relationships between dependent and independent variables. Learners will understand how to build, interpret, and evaluate linear and logistic regression models to make predictions and identify trends in data.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 2 books, 2 courses, 2 websites

[Learn Regression Analysis](/subject/regression-analysis)

### Statistical Software (R, Python)

This topic covers the use of R and Python for data analysis and visualization. Learners will write scripts to clean datasets, perform statistical tests, and generate data visualizations using libraries like pandas, ggplot2, and matplotlib.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 2 books, 5 websites

[Learn Statistical Software (R, Python)](/subject/statistical-software-r-python)

### Stochastic Calculus

Stochastic calculus handles calculus for random processes. You will learn Brownian motion, Itô's lemma, stochastic differential equations and the applications to option pricing and filtering.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 3 books, 1 websites

[Learn Stochastic Calculus](/subject/stochastic-calculus)

### Stochastic Processes

Stochastic processes are collections of random variables representing the evolution of a system over time. Learners will understand how to model and analyze systems governed by probabilistic laws, including random walks, Poisson processes, and continuous-time systems.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 2 books, 1 courses

[Learn Stochastic Processes](/subject/stochastic-processes)

### Variance & Covariance

Variance measures the spread of a single dataset, while covariance measures how two variables change together. Learners will understand how to calculate these metrics, interpret joint variability, and apply them to risk assessment and statistical modeling.

**Difficulty:** Beginner, Intermediate, Advanced

**Resources:** 1 videos, 1 websites, 1 papers

[Learn Variance & Covariance](/subject/variance-and-covariance)

---

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