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Suppose you want to calculate the likelihood of tossing two coins and coming up with one head. The great 18th-century mathematician Jean Le Rond d’Alembert thought the answer was obvious: there are three possibilities, zero, one or two heads. So the odds for any one of those happening must be one in three.

But as Leonard Mlodinow explains in “The Drunkard’s Walk: How Randomness Rules Our Lives,” there are, in fact, four possible outcomes: heads-heads, heads-tails, tails-heads and tails-tails. So there is a 25 percent chance of throwing zero or two heads and a 50 percent chance of throwing just one. In the long run, anyone offering d’Alembert’s odds in a coin-flipping contest would lose his shirt.


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