Games and gambling have been part of human cultures around the world for millennia. Nowadays, the connection between games of chance and mathematics (in particular probability) are so obvious that it is taught to school children. However, the mathematics of games and gambling only started to formally develop in the 17th century with the works of multiple mathematicians such as Fermat and Pascal. It is then no wonder that many incorrect beliefs around gambling have formed that are "intuitive" from a layman's perspective but fail to pass muster when applying the rigor of mathematics.
In this post, I want to discuss how surprisingly easy it is to be fooled into the wrong line of thinking even when approaching it using mathematics. We'll take a look from both a theoretical (mathematics) point of view looking at topics such as the Gambler's Fallacy and the law of small numbers as well as do some simulations using code to gain some insight into the problem. This post was inspired by a paper I recently came across a paper by Miller and Sanjurjo that explains the surprising result of how easily we can be fooled.