February 24, 2014 1:32:21 PM
Years ago my wife was looking through the books at a Goodwill store, and for a gag gift for my birthday she bought me a volume of poems by Rod McKuen, whose work I do not like. Helen brought it home and was going to wrap it when she looked inside. There was a dedication in the book signed by the author himself: "To Rob - May you always sleep warm. --Rod McKuen." Now I have a volume of McKuen's poetry I cannot throw away, but more to the point, how is it possible that completely by chance that she should pick up such a book bearing a dedication to someone of the same name? It just does not seem that such a thing could happen. It's the sort of story told many times in The Improbability Principle: Why Coincidences, Miracles, and Rare Events Happen Every Day (Scientific American / Farrar, Straus and Giroux) by mathematician David Hand. Hand knows that it is tough to write about mathematics for the layman, perhaps even more so for probability which he says "is renowned for its counterintuitive nature, more than any other area of mathematics. Even the most eminent mathematicians have been tripped up by it." Yet there are few equations here, there are many astounding stories, and many of Hand's explanations are drawn from tossing a humble die. (He collects dice, and has some that are weighted for cheating or have improper pips on the sides, and some that have ten sides, which if you know that there is no ten-sided regular polyhedron is a bit of a puzzle, but he will explain this to you.)
The book starts with its main contradiction, only a seeming contradiction and one of many explained away within these pages: How is it possible that extremely unlikely things happen, and not only happen, but happen over and over? Hand gives satisfying answers, but besides being a book about extraordinarily improbable events, this is also a book that explains probability in general. There was a time when we did not know that the Earth went around the Sun, and there was a time when we did not know that, say, coin tosses could be predicted in mass, even if we have no idea how an individual toss will come out. That we started to come to this understanding, Hand says in his brief history of probability, in the seventeenth century is yet another contradiction. Scientists like Newton and Boyle were showing how the universe was intrinsically deterministic, a clockwork universe with little role for chance. The impetus of trying to understand physical laws by the use of numbers, however, would have also favored a quantitative understanding of chance outcomes. The history lesson here winds up in a chapter titled "Life, the Universe, and Everything," explaining how chance affects evolution and even our existence in a cosmos that seems miraculously tuned just right for us.
Miracles, Hand reminds us repeatedly, happen all the time. We do try to seek patterns and explanations, though, and if we cannot understand the causes behind events, we have tended to attribute them to supernatural forces. Superstitions help us think we understand; if you go to a casino, you will see many gamblers who think, for instance, that blowing on the dice will give the cubes an extra impetus to fall as the thrower wants them to. If you believe in superstitions, you are likely to believe that leaves in a teacup or planets in the sky can be used to predict the future. (Hand reminds us that President Reagan's chief of staff has written that every major decision during his time there was cleared by an astrologer in San Francisco.) Miracles used to be the rightful realm of the gods who pulled the strings and made things turn out just so. Hand writes, "A little thought shows that it is in fact a useless explanation; it's just too powerful, since it can explain everything." To say, "The gods did it" and stop there explains nothing, but of course you will hear lottery winners declare that their particular god forced the numbers to go their particular way, while losers seldom claim that their god arranged the loss.
More satisfying is examining how probability holds sway in what we used to call miracles, and there are different branches of the improbability principle that Hand wants us to understand. Take the law of inevitability: Something has to happen. In a lottery, each possible ticket has so tiny a chance of winning that you might think it a miracle if yours is the one chosen. But what is certain is that some ticket will be picked to win (or if not, the stakes will be raised for the next draw when there will be a winner). It is thus dead certain that an improbable event will occur. There's the law of truly large numbers: if there are enough possible opportunities, any outrageously unlikely event can happen. If you toss a coin enough times, and have a near-eternity to do so, you will get a run of a hundred heads; it simply has to happen. These are understood strictly with probability theory, but we have also to supplement our "understanding" of rare events by human foibles. The law of selection, for instance, says that you can make probabilities as high as you like, in retrospect. It's like shooting arrows into the side of a barn and then painting bullseyes around each one. All of us are liable, too, to confirmation bias; we notice events that reinforce what we wish to believe and we disregard data that does not fit. Prophets and astrologers harness this tendency all the time. And sometimes we use bad equipment for our research. Federal law has specific and strict rules for every die thrown in a casino, but dice you get in a Monopoly set are far from such strict engineering and they have bias; research into psychokinesis, the ability to control die tosses, has been criticized because it used ordinary dice.
Indeed, many of Hand's illustrations come from the colorful history of the attempts to prove extrasensory perception and other paranormal abilities. The lab tests often involve the evaluation of subjects' abilities to know the outcome of a tossed coin beforehand or to influence the toss somehow, or some other manifestation of something better than chance. Hand gets to show how such experimenters cherry-pick data or willfully manipulate it. Replicable, careful studies have sometimes, and far from always, shown results that were slightly better than chance. If ESP works, it doesn't work all that often, and certainly not in any way that would be reliable or useful.
Readers of Hand's book will have a happy tour of many aspects of probability, delivered by a guide who is knowledgeable and funny. Even if some of the math gets by you, the astonishing stories are sure to impress you, like the ones about people who have won the lottery more than once. Spare a little sympathy, however, for Maureen Wilcox, who in 1980 bought lottery tickets in both the states of Massachusetts and Rhode Island. She picked all the right numbers, too. Except the numbers she picked for Rhode Island won in Massachusetts, and vice versa.