Richard’s Paradox demonstrated that a simple rule to define a set of numbers may lead to a paradox. Predecessors to the Richard Paradox In 1905, French mathematician Jules Richard shifted the focus from certain earlier mathematical paradoxes by showing that the definitions themselves might be at fault. In the very early 1900s, paradoxes in the […]

## Conditional Probability is Not Commutative: Formulas and Examples

Confusing the “given” event (the event that you assume to have occurred) with the combined event (for which you are calculating probability) is a common pitfall with conditional probability. A Recap of Conditional Probability Recall that “the conditional probability of event ‘A’, given that event ‘B’ has occurred, is calculated as the probability that both […]

## An Introduction to Calculating Conditional Probability in Mathematics

Despite the value of knowing the probability of an event before it occurs, it can be even more valuable to know how learning part of the outcome changes the conditional probability. The Foundation for Understanding Conditional Probability This article continues a series about probability, by introducing “conditional probability.” If the terms are unfamiliar, consider reviewing […]

## A Taste of the 2012 Joint Mathematics Awards and Prizes

The January 2012 Joint Mathematics Meetings featured an awards presentation in recognition of many outstanding mathematicians, educators and authors. The prizes are awarded by the AMS (American Mathematical Society), the Mathematical Association of America, the Association for Women in Mathematics, and the Society for Industrial and Applied Mathematics. With 33 individual recipients of 19 awards, certificates […]

## Axioms and Two Useful Theorems of Discrete Probability Functions

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

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 […]

## Power Sets come in Small, Infinite and Even Larger Sizes

## 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

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. […]