Axioms and Two Useful Theorems of Discrete Probability Functions

probability and dice

The first article in this series, Introducing Probability Theory without Statistics, noted that probability distribution might be “discrete” or “continuous.” This article builds the foundation for discrete probability functions, by introducing the four axioms and deriving two useful theorems from them. Discrete Probability Functions: The Soul of Discretion The phrase, “probability distribution”, refers to the […]

The Probability of the Allais Paradox in Lottery Preferences

"Taiwan Lottery Dream Sheet" by Prince Roy

One marvelous example of the conflict between mathematics and human behaviour is shown in the “Allais Paradox.” Compared to probability theory, in the Allais Paradox, people choose correctly or incorrectly based on irrelevant details. Probability, Payout, Expected Value and Lotteries The mathematical view of “probability” is the likelihood that some specific outcome will occur from […]

Introducing Probability Theory without Statistics


This article introduces basic mathematical concepts in probability. Future articles will discuss different aspects, including several paradoxical situations involving probabilities. For those who can’t wait, Solve the Monty Hall Problem using Logic and Mathematics. Probability, Statistics or Likelihood? In mathematics, “probability” is the study of how likely it is for some specific outcome to occur […]

Four Personalized Prime Number Formulae


The recent article, “Complex Tale of Eisenstein Prime Numbers“, was devoted to the prime numbers found by Ferdinand Gotthold Max Eisenstein. The names of several other mathematicians have become associated with their own sets of primes numbers. This article will introduce some of these very personalized primes. Fermat Numbers and Fermat Primes Pierre de Fermat […]

The Complex Tale of Eisenstein Prime Numbers

Eisenstein Prime Numbers and Units - Image by Januszkaja

Last week’s “Several Different Paths to Prime Numbers” opened with this intriguing image. Unfortunately, there was no room to answer the question “What are Eisenstein primes?” An Explanation of an Eisenstein Prime Number We already know that a “prime number” is a number that can only be evenly divided by itself and the number one. […]

Several Different Paths to Prime Numbers

Visual representation of Eisenstein Primes: Image by Johannes Rössel

Last week’s “Brief Introduction to Prime Numbers” dangled a few teasers – since keen minds are eager to know more, let’s tie up some of the loose ends. Pure Review: What is a Prime Number? A “Prime” number is a natural number greater than one, that is only evenly divisible by itself and one. This […]

Filtering Prime Numbers using the Sieve of Eratosthenes

sieve of eratosthenes

What is the Sieve of Eratosthenes? Last week’s article, A Brief Introduction to Prime Numbers, mentioned the “Sieve of Eratosthenes” – a procedure devised by the clever Greek philosopher Eratosthenes. As a sieve catches fish, but allows water to escape, the Sieve of Eratosthenes retains prime numbers but allows composite numbers to pass through. Essentially, […]

A Brief Introduction to Prime Numbers

"Sieve for Seven" by Scot Nelson

Would you like a simple introduction to primes? This article introduces a series about prime numbers by answering the following basic questions: What are prime numbers? Are primes important? How many prime numbers are there? What is a Prime Number? Prime numbers are those Natural numbers that are greater than one and can only be […]

The Definitive Quick Reference Guide to All Types of Numbers

"Aleph" by Renaud Camus

Background to this Reference Guide to Numbers Several previous articles introduced a variety of numbers, including Natural Numbers, Integers, Rational Numbers, Real Numbers, Imaginary Numbers, Irrational Numbers, Infinite Numbers, and others. This quick reference guide is intended to provide additional information and a summary of the main types of numbers. The Sets of Numbers The […]