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Driving on a hot summer's day, you might notice a pool of water shimmering on the asphalt a few hundred feet ahead. By the time you reach the spot, it will have vanished, perhaps to appear again a bit farther along. It's an optical illusion, of course, caused by light reflecting off a layer of very hot air above the road top. And in remembering that fact, you know better than to trust the evidence of your eyes.

Yet you continue seeing it: water in the distance, visible to the same brain capable of understanding that it isn't actually there. Knowing something to be a mirage is not enough to make it disappear.

Likewise with the conceptual phantom Barbara Blatchley considers in What Are the Chances? Why We Believe in Luck (Columbia University Press). A professor of psychology and neuroscience at Agnes Scott College, Blatchley is also the author of a recent intro-to-statistics textbook that has been praised for its clarity. Parts of her new book build on the earlier one: beliefs about winning or losing streaks, the efficacy of superstitious practices and so forth are to some degree manifestations of a shaky grasp of the principles of probability. (An awful lot of human experience ultimately derives from people being bad at math.) But more than that is at play in the persistent illusion of luck's fickle operations.

Suppose you have flipped a coin, and it has come up heads nine times in a row. A suspicion that the coin has been rigged is understandable, perhaps even warranted, but for the sake of this thought experiment (adapted from Blatchley's opening pages), let's assume things are on the level. The odds of a flipped coin coming up heads 10 times in a row are one in 1,024. Anyone watching and/or wagering will almost certainly expect the 10th flip to come up tails, with a gut feeling that the chances of that outcome being something on the order of 99.9 percent (i.e., 1,023 out of 1,024).

Such is the kind of thinking that ensures only the most spectacularly incompetent of casino owners will ever go broke. In reality, the chances of getting heads or tails on the 10th flip is exactly the same as it has been on each of the first nine: 50 percent. True, the odds of getting the same outcome 10 times in a row are minuscule -- but each flip is a discrete event, with nothing in the first nine having the slightest bearing on the outcome of the 10th. The universe is not keeping track. This is logically irrefutable but somehow counterintuitive for most of us, it seems.

"Randomness," Blatchley writes, "can and does happen in streaks and clusters. It’s humans who tend to say that random events should not form a pattern, and when they do, we say they can’t be random anymore." She quotes two experimental psychologists: "Few ideas are so deeply engraved in our minds as the notion that events have their causes. The notion of causeless-ness is so alien to us that in the absence of a known cause, we tend to attribute events to imaginary causes." Luck can be personified (as the Greek goddess Tyche or the Hindu deity Ganesha) or reified as a force that can be controlled through superstitious means (e.g., talismans or rituals, traditional or personal). But at perhaps the most basic level, it operates as the brain's ad hoc response to a world neither completely orderly nor totally chaotic.

While improvised and usually inadequate, these sense-making efforts are not themselves random. The author surveys research showing that our tendency to attribute causality fall into patterns that derive from our experience in dealing with other people. When considering the behavior of others, "we tend to overestimate the effect of personality and disposition. Other people act the way they do because they're just that kind of people." But in assessing ourselves, "we overestimate the influence of the situation and underestimate the influence of our own characteristics. After all, we know what kind of people we are, and we're right in the middle of the situation, so we often look to the external environment to explain what we're doing and why we're doing it." When an outcome seems to derive from neither objective circumstances nor an individual's own capacities or characteristics, up comes the sense that something else must be involved -- luck, good or bad, experienced as an invisible factor in the world or as something the individual "has."

The ability to imagine alternative outcomes -- the human capacity for counterfactual thinking -- also refracts our experience and provokes efforts to understand why one possibility came to pass and another did not. The level of anxiety typical of a given nervous system also has an effect on how much randomness or unpredictability the individual can handle. Which is not to say that thoughts about luck are necessarily symptomatic. A belief in luck -- or in the possibility of attracting the good variety -- may have a degree of therapeutic benefit. It is not always the worst of illusions.