Throughout times people have often shown a great interest in games, especially those that includes money.

Gradually, people started to ask questions about something they had thought were unexpected and based on luck, such as:

• What is the probability to draw an ace and after that draw another ace?
• What is the probability of getting six twice when you toss a die just as many times?

Gamblers in the Middle Ages often raised questions about their chances to either win or lose a game. Even if some of these people got a huge experience and developed a solid intuition of what could pay, didn’t the first mathematical studies on probability appear until the 16^th century.

One of the earliest mathematical studies on probability was, "On Casting the Die". It was written by the 16^th century Italian mathematician and physician Cardano, but it was not published until 1663, 87 years after the death of Cardano. He introduced concepts of combinatorics into calculations of probability and defined probability as "the number of favorable outcomes divided by the number of possible outcomes.

In the 17^th century, questions about the probability of events occurring in games of chance were discussed in correspondence between the French mathematicians Pascal and de Fermat. Building of their results, the Dutch physicist-astronomer mathematician Huygens published, in 1656, "On Reasoning in Games of Chance".

In the late 18^th century, it became increasingly evident that analogies exist between games if chance and random phenomena in physical, biological, and social sciences.

Contributions of fundamental importance to probability theory were made in the latter half of the 18^th century and the beginning of the 19^th century. The most important publication on probability theory in this area is Laplace’s "Théorie analytique des probabilitès (1812), which discussed practical applications of the theory and developed the concepts of normal distribution, first discovered by Abraham de Moivre, an important person concerning the probability problems. In 1718 he published the book "Doctrine of Chances", which has been remembered as a pioneer work. He invented the gausscurve. Moivre used this curve to calculate probabilities in huge experiment-series with binomial experiments. Here is an example.

In the beginning of the 20^th century, the need for applications of probability theory increased in physics, economics, insurance, and telephone communication. Albert Einstein gave important impulses to this. Applications often precipitated new probability problems, which had to be tackled within the field of theoretical probability, and thus a fruitful interplay between the sciences was created.

The English geneticist and statistician R.A. Fisher was professor of genetics at Cambridge from 1943-57. He was one of most eminent scientists of the 20^th century, and developed methods of multivariate analysis – analysis of problems involving more than one variable – and used them in his investigations of linkage of genes to various traits. He also introduced the idea of likelihood in statistical inference that is, how to draw conclusions on the basis of the relative probability of different events.

The theory of games was founded by John von Neumann, preeminent 20^th century innovator in many fields of pure and applied mathematics. He created a mathematical model for games of chance, such as poker and bridge, that involve free choices-strategy – for the players. His first paper on this subject was presented in 1926. Von Neumann’s theories were further developed in his major work "Theory of games and Economic Behavior, co-authored with the economist Oskar Morgenstern. The theory of games is now a mathematical discipline of it’s own with far-reaching applications to economics and social sciences.

As previously mentioned, the French mathematician Pierre-Simon Laplace published "Théorie analytique des probabilités", but he also presented the first description of probability in 1795. He said that: "If an experiment can produce a number of different and equally outcomes, some of which can be considered favorable. Then the probability of a favorable outcome is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes." The most familiar example of such an experiment is the tossing of a coin. Provided the coin is fair, the probability of success will always be ½.

Later it was discovered that Laplace’s description of probability couldn’t be considered a mathematical definition because it depends on the mathematical idea of "equally likely outcomes". But it can be used as a guide in the calculation of many simple problems. An acceptable mathematical definition was not formulated until the 20^th century.

In the early contributions to mathematical probability theory, primarily those of French mathematicians Blaise Pascal and Pierre de Fermat and the Dutch mathematician, astronomer, and physicist Christian Huygens which all lived in the 17^th century. Their main consideration was the expected gain in games of chance and considerations other than the probability itself. In Huygens’ treatise "De Ratiociniis in Ludo Aleae" ("On Ratiocination in Dice Games") appeared the first documented reference to the concept of mathematical expectation.

As an illustration of the use of products of probabilities, Huygens’ famous "fifth problem" will be considered. This problem evolved trough various generalizations and extensions into the problem of gambler’s ruin, duration of play, simple random walk, and many others that have occupied many mathematicians on different occasions, from the time of Huygens to the middle of the 20^th century. In the its simplest original form, it is as follows:

The usual explanation of probability today, is that the probability for a result is the limit-value for the relative frequency for this result when the experiment is continually repeated.

As we walk into the 21^th century probability is used in many fields. If you are buying a house you might be interested in whether the price will be reduced or not. Or if you want to lend money and want to know the probability of a reduction in the interest. If you a critical disease and wonder what the odds are for surviving.

As you can see probability can be quite useful.

This article is written by:

• Stine Holstangen
• Hege Øren
• Kristian Kolstad

Greåker V.G.S., January 22, 1999