At the beginning of this book, I noted that some aspects of probability appear, at first sight, to defy common sense. Examples have arisen as the story has unfolded. Here are some other circumstances where intuition can be misleading, but, with sufficient care, these apparent contradictions can be explained. The subject of probability is wholly free from real paradoxes. But although ideas of probability can help us make sensible decisions, we may also find that even thinking about the probabilities of certain events might lead to uncomfortable dilemmas.
Parrondo’s paradox
Graham Greene’s novel Loser Takes All is a fine read, but based on a false premise: that there is some clever mathematical way of combining bets on a roulette wheel to give the player an advantage, rather than the house. On the contrary: mathematics has proved that, when all individual bets favour the house, no combination whatsoever can turn matters round and favour the player. Sorry, folks.
Juan Parrondo has shown that you have to be very precise in how you formulate a general claim that, whenever all bets favour one side, it is impossible to combine bets so that the other side has an advantage. I describe here a modification of his idea, due to Dean Astumian, who described a simple game played on the board with five slots, shown in Figure 11 . (This is not a serious game. It was constructed merely to make this point.)
11. The board for Astumian’s game
You need some way of generating a random event that will occur 1% of the time: perhaps a bag with 99 White balls and one Black ball, or a spinner that is equally likely to come to rest on any one of its one hundred sides. To begin the game, place a token on the slot marked ‘Start’. Every move will take the token one step left, or one step right, and you win if the token reaches Win before it hits Lose.
There are two basic sets of rules, call them Andy and Bert. With Andy, from Start you always move to Left, and from Right, you always move to Win. From Left, you use the spinner, to give a 1% chance of moving to Lose, and a 99% chance of moving back to Start. With Bert, the spinner is used for Start to give a 99% chance of moving to Right, and a 1% chance of going to Left. From Right,