Mathematics for Game Development

9.

Probability and Randomness

9.1.

Probability Theory

9.1.1.

Basic Probability Concepts

9.1.1.1.

9.1.1.2.

9.1.1.3.

Probability Measures

9.1.2.

Probability Rules

9.1.2.1.

9.1.2.2.

Multiplication Rule

9.1.2.3.

Complement Rule

9.1.3.

Conditional Probability

9.1.3.1.

9.1.3.2.

9.1.4.

Random Variables

9.1.4.1.

Discrete Random Variables

9.1.4.2.

Continuous Random Variables

9.1.4.3.

9.1.4.4.

9.2.

Random Number Generation

9.2.1.

Pseudorandom Number Generators

9.2.1.1.

Linear Congruential Generators

9.2.1.2.

Mersenne Twister

9.2.1.3.

Xorshift Generators

9.2.2.

True Random Number Generation

9.2.2.1.

Hardware Sources

9.2.2.2.

Environmental Noise

9.2.3.

Random Number Quality

9.2.3.1.

9.2.3.2.

9.2.3.3.

9.2.4.

Seeding and Reproducibility

9.2.4.1.

9.2.4.2.

Deterministic Sequences

9.2.4.3.

Save State Compatibility

9.3.

Probability Distributions

9.3.1.

Uniform Distribution

9.3.1.1.

Discrete Uniform

9.3.1.2.

Continuous Uniform

9.3.1.3.

Generation Methods

9.3.2.

Normal Distribution

9.3.2.1.

Gaussian Properties

9.3.2.2.

Box-Muller Transform

9.3.2.3.

Central Limit Theorem

9.3.3.

Other Distributions

9.3.3.1.

Exponential Distribution

9.3.3.2.

Poisson Distribution

9.3.3.3.

Binomial Distribution

9.3.4.

Distribution Sampling

9.3.4.1.

Inverse Transform Method

9.3.4.2.

Rejection Sampling

9.3.4.3.

Importance Sampling

9.4.

Game Applications

9.4.1.

Loot and Reward Systems

9.4.1.1.

Drop Rate Tables

9.4.1.2.

Weighted Random Selection

9.4.1.3.

Rare Item Generation

9.4.1.4.

Anti-Frustration Mechanics

9.4.2.

Procedural Content Generation

9.4.2.1.

Random Map Generation

9.4.2.2.

Terrain Generation

9.4.2.3.

9.4.2.4.

9.4.3.

9.4.3.1.

Decision Making

9.4.3.2.

State Transitions

9.4.3.3.

Behavioral Variation

9.4.3.4.

Emergent Behavior

9.4.4.

9.4.4.1.

Hit Chance Calculations

9.4.4.2.

Critical Hit Systems

9.4.4.3.

Damage Variation

9.4.4.4.

Status Effect Probability

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10. Curves and Splines