Math Visualization List
Calculus
First Derivative Second Derivative Integration
Determinant
2x2 & 3x3 Determinants Cramer's Rule Permutations & Inversions Permutation Definition & Expansion Properties of Determinants: Row Operations Minors & Cofactors Two-Row Laplace Expansion Vandermonde Determinant
Matrix Operations
Matrix Addition, Subtraction & Scalar Multiplication Matrix Multiplication & Power Transpose, Determinant & Adjoint Inverse Matrix & Matrix Equations Elementary Transformations, Echelon & Canonical Form
Probability & Statistics
Conditional Probability, Independence & Total Probability Discrete Probability Distributions PMF & CDF 2D Random Vectors Conditional Probability Distributions Bayes' Theorem Numerical Characteristics of Random Variables Higher Moments & Covariance Matrix Law of Iterated Expectations Independence, Mean Independence & Uncorrelated Continuous Statistical Distributions
Vector Operations
Vector Addition, Subtraction & Scalar Multiplication Linear Combination, Span & Basis Matrices as Transformations Composition of Transformations 3D Linear Transformations Determinant Inverse, Column Space & Rank Cross-Dimensional Transformations Vector Angle Cross Product Change of Basis Eigenvectors & Eigenvalues Function Vector Spaces Geometric Interpretation of Cramer's Rule Gravity Simulation

Probability Theory Visualization

Interactive Exploration of Conditional Probability, Independence & Total Probability

🎯 Select Demo Mode

⚙️ Parameter Controls

📊 Probability Data

Conditional Probability Formula:
P(A|B) = =
When event B occurs, the probability of A is the proportion of the intersection within B
Total Probability Formula:
P(A) = + =
The total probability of event A is the weighted sum of probabilities through all possible paths

📌 Observation: Adjust the sliders to change P(A), P(B), and P(A∩B). When independent events is enabled, P(A∩B) is automatically calculated as P(A)×P(B). The blue circle represents B, the red circle represents A, and the purple overlapping area represents A∩B. Conditional probability P(A|B) is the proportion of the purple area within the blue circle.

📌 Observation: This is a probability decision tree. B₁ and B₂ are mutually exclusive and exhaustive events (causes), and A is the result. The probability of each path is the product of conditional probability and prior probability. P(A) is the sum of probabilities of all paths that lead to A.

📖 Total Probability Formula Derivation

Step 1: Understand the Premise

设事件组 B₁, B₂, ..., Bₙ 是样本空间 Ω 的一个划分（即 Bᵢ 互不相容且 ∪Bᵢ = Ω）

对于任意事件 A，我们有：

Step 2: Decompose Event A

A = A ∩ Ω = A ∩ (∪Bᵢ) = ∪(A ∩ Bᵢ)

由于 Bᵢ 互不相容，则 A ∩ Bᵢ 也互不相容

Step 3: Apply Countable Additivity

P(A) = P(∪(A ∩ Bᵢ)) = ∑P(A ∩ Bᵢ)

Step 4: Apply Multiplication Rule

对每一项使用乘法公式：P(A ∩ Bᵢ) = P(A|Bᵢ)P(Bᵢ)

最终公式：

P(A) = ∑ P(A|Bᵢ)P(Bᵢ)

其中 i = 1, 2, ..., n

🔄 Animation Controls

Step: /5