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Probabilities and likelihoods are everywhere around us, from weather forecasting to games, insurance or election polls. However, in the history of mathematics, probability is actually a very recent idea. While geometry and algebra were studied by ancient Greek mathematicians more than 2500 years ago, the concepts of probability only emerged in the 17th and 18th century.

According to legend, two of the greatest mathematicians, Blaise Pascal and Pierre de Fermat, would regularly meet up in a small cafe in Paris.

To distract from the difficult mathematical theories they were discussing, they often played a simple game: they repeatedly tossed a coin – every heads was a point for Pascal and every tails was a point for Fermat. Whoever had fewer points after three coin tosses had to pay the bill.

One day, however, they get interrupted after the first coin toss and Fermat has to leave urgently. Later, they wonder who should pay the bill, or if there is a fair way to split it. The first coin landed heads (a point for Pascal), so maybe Fermat should pay everything. However, there is a small chance that Fermat could have still won if the two next tossesnext coin toss would have been tails.

Pascal and Fermat decided to write down all possible ways the game could have continued:


Pascal wins


Pascal wins


Pascal wins


Fermat wins

All four possible outcomes are equally likely, and Pascal wins in threefourtwo of them. Thus they decided that Fermat should pay 3/4 of the bill and Pascal should pay 1/4.

Pascal and Fermat had discovered the first important equation of probability: if an experiment has multiple possible outcomes which are all equally likely, then

Probability of an event = Number of ways the event could happenTotal number of possible outcomes.

In our example, the probability of Pascal winning the game is 34=0.75, and the probability of Fermat winning the game is 14=0.25.