Probability theory seeks to apply mathematical rules to the analysis of random events. It aims to model and predict likely outcomes under uncertain conditions. It also evaluates the likelihood of a particular hypothesis occurring on the basis of the available evidence and taking into consideration the impact of stochastic processes – that is to say, processes that may be analysed statistically but not predicted precisely.
Gambling on probability theory
In 16th-century Italy, the polymath and keen games-player, Gerolamo Cardano, first seriously attempted to use mathematics to predict the outcomes of games of chance. Then, in the following century, Pierre de Fermat and Blaise Pascal conducted a legendary correspondence on topics such as how best to divide a stake in a game of chance. Around the same time, Christiaan Huygens wrote the most comprehensive study of the subject thus far.
Take, say, a simple game of coin toss. Using a standard, unadulterated coin, it is obvious that there is a 50-50 chance of the coin landing on heads, and the same probability of tails. Furthermore, given no skulduggery, it is impossible to predict accurately what the result will be. However, by looking at a larger statistical sample, certain patterns can be traced. The tossing of a coin is therefore a stochastic process – unpredictable at the level of the individual instance but more predictable on a larger scale. The law of large numbers, for example, says that if an event with the same likelihood of separate outcomes (such as a coin-toss) is carried out enough times, the occurrence of each particular outcome will even out. So if you toss a coin once or twice, you cannot predict the likely ratio of heads-to-tails results. However, if you toss the coin a thousand times, you might reasonably expect close to 500 instances of tails, and 500 of heads.
The Swiss Jacob Bernoulli (1654–1705) and the French Pierre-Simon Laplace (1749–1827) put probability theory on a still sounder mathematical footing before the Russian Andrei Markov (1856–1922) ushered in the period of modern probability theory – not least through his elucidation of ‘memorylessness’, whereby future states may be predicted only on the evidence of the present state and not past events. His fellow countryman, Andrei Kolmogorov (1903–87), further built upon and extended Markov’s insights over the course of the 20th century.
Probability theory has many obvious applications where there is gain to be made from accurate forecasting – from gaming and financial speculation to meteorology and insurance. But, arguably, probability theory’s most profound influence is in a field undreamt of by Cardano and the other early pioneers: quantum mechanics, the study of the sub-atomic world that abounds in apparent unpredictability.