Concepts35
Category
Sprague-Grundy Theorem
Sprague–Grundy theory turns every finite impartial game (normal play) into an equivalent Nim heap with a size called the Grundy number.
Game Theory - Nim
Nim is a two-player impartial game with several piles where a move removes any positive number of stones from exactly one pile.
Game Theory - Calculation Techniques
Sprague–Grundy theory converts any impartial, normal-play game into an equivalent Nim heap using a Grundy number.
Variance and Covariance
Variance measures how spread out a random variable is around its mean, while covariance measures how two variables move together.
Linearity of Expectation Applications
Linearity of expectation says the expected value of a sum equals the sum of expected values, even if the variables are dependent.
Expected Value
Expected value is the long-run average outcome of a random variable if you could repeat the experiment many times.
Bayes' Theorem
Bayes' Theorem tells you how to update the probability of a hypothesis after seeing new evidence.
Probability Fundamentals
Probability quantifies uncertainty by assigning numbers between 0 and 1 to events in a sample space.
Catalan Numbers
Catalan numbers count many 'non-crossing' and 'well-formed' structures like balanced parentheses, binary trees, Dyck paths, and triangulations of a convex polygon.
Derangements
A derangement is a permutation with no element left in its original position, often written as !n or D(n).
Lucas' Theorem
Lucas' Theorem lets you compute C(n, k) modulo a prime p by working digit-by-digit in base p.
Inclusion-Exclusion Principle
The Inclusion-Exclusion Principle (IEP) corrects overcounting by alternately adding and subtracting sizes of intersections of sets.