Probability distribution

A probability distribution is a function that maps each outcome in a sample space to its probability.

For a probability distribution $p$ over a countable sample space $S$, it is a rule that: $$\sum\limits_{s \in S}p(s) = 1$$

Summing up the probabilities of every single possible outcome in an experiment should be $1$, because the whole point of an experiment is to be guaranteed that an outcome from a sample space will occur.

Just for fun, imagine an unfair coin where heads is $3$ times more likely to come up than tails. Then, $p(H) = \frac{3}{4}$ and $p(T) = \frac{1}{4}.$ Adding up the probabilities of every outcome in the sample space $\set{H, T}$ results in $\frac{3}{4} + \frac{1}{4} = 1$, as expected.

Let's also consider a biased die where $1$ is $5$ times more likely to be rolled than any other number. Then, $p(1) = \frac{5}{10}$ and $p(2) = p(3) = p(4) = p(5) = p(6) = \frac{1}{10}.$ The probability of the event of rolling an odd number occurring is simply the sum of all the probabilities of the outcomes that are odd numbers (all the elements of that event): $p(1) + p(3) + p(5) = \frac{5}{10} + \frac{1}{10} + \frac{1}{10} = \frac{7}{10}.$

Logic & Proofs

Integer • Rational number • Inequality • Real number • Theorem • Proof • Statement • Proof by exhaustion • Universal generalization • Counterexample • Existence proof • Existential instantiation • Axiom • Logic • Truth • Proposition • Compound proposition • Logical operation • Logical equivalence • Tautology • Contradiction • Logic law • Predicate • Domain • Quantifier • Argument • Rule of inference • Logical proof • Direct proof • Proof by contrapositive • Irrational number • Proof by contradiction • Proof by cases • Summation • Disjunctive normal form

Set Theory

Set • Element • Empty set • Universal set • Subset • Power set • Cartesian product • String • Binary string • Empty string • Set operation • Set identity • Set proof

Functions

Function • Floor function • Ceiling function • Inverse function

Algorithms

Algorithm • Pseudocode • Command • Asymptotic notation • Time complexity • Atomic operation • Brute-force algorithm

Relations

Relation • Reflexive relation • Symmetric relation • Transitive relation • Relation composition • Equivalence relation • Equivalence class

Number Theory

Integer division • Linear combination • Division algorithm • Modular arithmetic • Prime factorization • Greatest common divisor • Least common multiple • Primality test • Factoring algorithm • Euclid's theorem • Prime number theorem • Euclidean algorithm

Induction

Proof by induction • Fibonacci sequence • Proof by strong induction • Well-ordering principle • Sequence • Factorial • Recursive definition

Combinatorics

Rule of product • Rule of sum • Bijection rule • Permutation • Combination • Complement rule • Experiment • Outcome • Sample space • Event • Probability • Probability distribution • Uniform distribution • Multiset • Sixfold way • Inclusion-exclusion principle • Pigeonhole principle

Graph Theory

Graph • Walk • Subgraph • Regular graph • Complete graph • Empty graph • Cycle graph • Hypercube graph • Bipartite graph • Component • Eulerian circuit • Eulerian trail • Hamiltonian cycle • Hamiltonian path • Tree • Huffman tree • Substring • Forest • Path graph • Star • Spanning tree • Weighted graph • Minimum spanning tree • Greedy algorithm • Prim's algorithm

Recursion

Recursion • Recursive algorithm • Correctness proof • Divide-and-conquer algorithm • Sorting algorithm • Merge sort