Is God a Mathematician_ - Mario Livio [65]
As some of the early statisticians readily realized, both the collection of statistical data and their interpretation can be very tricky and should be handled with the utmost care. A fisherman who uses a net with holes that are ten inches on a side might be tempted to conclude that all fish are larger than ten inches, simply because the smaller ones would escape from his net. This is an example of selection effects—biases introduced in the results due to either the apparatus used for collecting the data or the methodology used to analyze them. Sampling presents another problem. For instance, modern opinion polls usually interview no more than a few thousand people. How can the pollsters be sure that the views expressed by members of this sample correctly represent the opinions of hundreds of millions? Another point to realize is that correlation does not necessarily imply causation. The sales of new toasters may be on the rise at the same time that audiences at concerts of classical music increase, but this does not mean that the presence of a new toaster at home enhances musical appreciation. Rather, both effects may be caused by an improvement in the economy.
In spite of these important caveats, statistics have become one of the most effective instruments in modern society, literally putting the “science” into the social sciences. But why do statistics work at all? The answer is given by the mathematics of probability, which reigns over many facets of modern life. Engineers trying to decide which safety mechanisms to install into the Crew Exploration Vehicle for astronauts, particle physicists analyzing results of accelerator experiments, psychologists rating children in IQ tests, drug companies evaluating the efficacy of new medications, and geneticists studying human heredity all have to use the mathematical theory of probability.
Games of Chance
The serious study of probability started from very modest beginnings—attempts by gamblers to adjust their bets to the odds of success. In particular, in the middle of the seventeenth century, a French nobleman—the Chevalier de Méré—who was also a reputed gamester, addressed a series of questions about gambling to the famous French mathematician and philosopher Blaise Pascal (1623–62). The latter conducted in 1654 an extensive correspondence about these questions with the other great French mathematician of the time, Pierre de Fermat (1601–65). The theory of probability was essentially born in this correspondence.
Let’s examine one of the fascinating examples discussed by Pascal in a letter dated July 29, 1654. Imagine two noblemen engaged in a game involving the roll of a single die. Each player has put on the table thirty-two pistoles of gold. The first player chose the number 1, and the second chose the number 5. Each time the chosen number of one of the players turns up, that player gets one point. The winner is the first one to have three points. Suppose, however, that after the game has been played for some time, the number 1 has turned up twice (so that the player who had chosen that number has two points), while the number 5 has turned up only once (so the opponent has only one point). If, for whatever reason, the game has to be interrupted at that point, how should the sixty-four pistoles on the table be divided between the two players? Pascal and Fermat found the mathematically logical answer. If the player with two points were to win the next roll, the sixty-four pistoles would belong to him. If the other player were to win the next roll, each player would have had two points, and so each would have gotten thirty-two pistoles. Therefore, if the players separate without playing the next roll, the first player could correctly argue: “I am certain of thirty-two pistoles even if I lose this roll, and as for the other thirty-two pistoles perhaps I shall have them and perhaps you will have them; the chances are equal. Let us then divide these thirty-two pistoles equally and give me also the thirty-two pistoles of which I am certain.” In other words, the first player