Gambling, probability and Bernoulli trials
It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge.
Pierre-Simon, Marquis de Laplace, Théorie Analytique des Probabilités, 1812
The developments of gambling and probability theory have gone hand in hand throughout history: humans have been playing dice and cards for centuries, humans have then started formalising strategies for wins and losses.
Primitive forms of dice are among the first playthings invented by humanity; card games were likely invented in ancient China in the 9th century. There are references to games in numerous pieces of literature, for instance the "Zara" dice game, from the Middle Ages, is mentioned in Dante Alighieri's Comedy and other works.
The Zara game is very simple: you throw three 6-sided dice at once after choosing a number 3-18 (respectively the minimum and maximum achievable) - because results in the middle of the scale are obtainable with a larger number of dice combinations (you get a 3 with 1-1-1 only, a 18 with 6-6-6 only, a 4 with 1-1-2/1-2-1/2-1-1, a 17 with 6-6-5, 6-5-6, 5-6-6, ..., but a 10 or 11 with 27 combos each, out of the 216 possible ones), obviously it's convenient to bet on them. It does feel like an uninteresting game because of this, but maybe they hadn't figured it out yet back then! This game is very useful as toy example to teach concepts in probability.
Starting from the 17th century a whole new area of mathematics developed to investigate and formalise the concepts and measurements around chance, odds, wins and losses. Fermat, Pascal, Huygens, Bernoulli, Cardano, de Moivre, Laplace are some of the famous names involved in the founding of this new intellectual endeavour. I'd love to know if the pioneers of probability were also gamblers themselves, but I'm not sure I found reliable information. Either way, it was during their times that playing and gambling took on a whole new level in Europe.
It is not uncommon anyway to have people involved in mathematics and forms of gambling or investing to this day - Jim Simons is a brilliant example: a mathematician by background and career with important academic contributions and prizes won, he also worked at the NSA and eventually launched a hedge fund company which made him a billionaire. He's also been a great philantropist who contributed all his life, monetarily and with time and energy to causes related to the dissemination of scientific education and research, establishing and funding, amongst other nonprofits, "Math for America".
We'll give a brief overview of something from the early era of probability theory: Bernoulli trials and the related probability distributions.
The Bernoulli distribution
Let's consider a binary variable X, that is, one that can take only 2 values, say "1" (usually representing success) with probability p and "0" (usually representing failure), with probability 1-p. Phenomena behaving like this are known as Bernoulli trials - examples are a coin flip or randomly asking people in the street whether they had cereal for breakfast or not.
We can write the probability mass function (the mathematical form of probability for each value) as
because when x=1 we are left with p and when x=0 with 1-p. This is the Bernoulli distribution.
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