A Panorama of Statistics

10 Puzzles and paradoxes in probability

As you may already have discovered, there are lots of puzzles and paradoxes in theoretical and applied probability. In this chapter, we want to look a little more deeply into why this subject is so rich in difficulties. First, though, it’s good to be clear about who finds work with probability difficult. The answer is … everybody. Even statisticians! And that has been so for a long time.

In 1654, the French mathematician Blaise Pascal engaged his great contemporary Pierre de Fermat in a joint inquiry on two fundamental matters: how to assign numerical probabilities to chance events, and how to determine the probabilities of compound events. These tasks had always seemed so challenging that no real start had been made on them since, many centuries earlier, a profound realisation had been written into the Bible (Ecclesiastes 9:11): ‘… time and chance happen to all [mankind]’. However, whereas time has been measured for some 4000 years (these days, with supreme precision and accuracy), efforts to measure chance are hardly 400 years old, and the concomitant problems are still not solved to everyone’s satisfaction. For a short overview of some of these problems, see Good (1959).

It is mostly the finer aspects of these unsolved problems that give rise to puzzles in probability. Probability theory is full of subtleties. If they are neglected or unrecognised, difficulties of understanding and interpretation soon ensue. There is another ...