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Thinking of future events in terms of probabilities is difficult enough for even coin tossing or weather forecasting, where more or less the same kind of thing is happening over and over again. But for events that happen only once in a lifetime, like the outbreak of a war, the election of a president, or even which college you get accepted to, the distinction becomes almost impossible to grasp. What does it mean, for example, to have said the day before Barack Obama’s victory in the 2008 presidential election that he had a 90 percent chance of winning? That he would have won nine out of ten attempts? Clearly not, as there will only ever be one election, and any attempt to repeat it—say in the next election—will not be comparable in the way that consecutive coin tosses are. So does it instead translate to the odds one ought to take in a gamble? That is, to win $10 if he is elected, I will have to bet $9, whereas if he loses, you can win $10 by betting only $1? But how are we to determine what the “correct” odds are, seeing as this gamble will only ever be resolved once? If the answer isn’t clear to you, you’re not alone—even mathematicians argue about what it means to assign a probability to a single event.13 So if even they have trouble wrapping their heads around the meaning of the statement that “the probability of rain tomorrow is 60 percent,” then it’s no surprise that the rest of us do as well.

The difficulty that we experience in trying to think about the future in terms of probabilities is the mirror image of our preference for explanations that account for known outcomes at the expense of alternative possibilities. As discussed in the previous chapter, when we look back in time, all we see is a sequence of events that happened. Yesterday it rained, two years ago Barack Obama was elected president of the United States, and so on. At some level, we understand that these events could have played out differently. But no matter how much we might remind ourselves that things might be other than they are, it remains the case that what actually happened, happened. Not 40 percent of the time or 60 percent of the time, but 100 percent of the time. It follows naturally, therefore, that when we think about the future, we care mostly about what will actually happen. To arrive at our prediction, we might contemplate a range of possible alternative futures, and maybe we even go as far as to determine that some of them are more likely than others. But at the end of the day, we know that only one such possible future will actually come to be, and we want to know which one that is.

The relationship between our view of the past and our view of the future is illustrated in the figure on the facing page, which shows the stock price of a fictitious company over time. Looking back in time from the present, one sees the history of the stock (the solid line), which naturally traces out a unique path. Looking forward, however, all we can say about the stock price is its probability of falling within a particular range. My Yahoo! colleagues David Pennock and Dan Reeves have actually built an application that generates pictures like this one by mining data on the prices of stock options. Because the value of an option depends on the price of the underlying stock, the prices at which various options are being traded now can be interpreted as predictions about the price of the stock on the date when the option is scheduled to mature. More precisely, one can use the option prices to infer various “probability envelopes” like those shown in the figure. For example, the inner envelope shows the range of prices within which the stock is likely to fall with a 20 percent probability, while the outer envelope shows the 60 percent probability range.

We also know, however, that at some later time, the stock price will have been revealed—as indicated by the dotted “future” trajectory. At that time, we know the hazy cloud of probabilities