What Are the Odds?

Consider:

“The true meaning of the word [odds] is ”a surprising concurrence of events, perceived as meaningfully related, with no apparent causal connection.” In other words, pure happenstance. Yet by merely noticing a coincidence, we elevate it to something that transcends its definition as pure chance. We are discomforted by the idea of a random universe. Like Mel Gibson’s character Graham Hess in M. Night Shyamalan’s new movie ”Signs,” we want to feel that our lives are governed by a grand plan. The need is especially strong in an age when paranoia runs rampant. ”Coincidence feels like a loss of control perhaps,” says John Allen Paulos, a professor of mathematics at Temple University and the author of ”Innumeracy,” the improbable best seller about how Americans don’t understand numbers. Finding a reason or a pattern where none actually exists ”makes it less frightening,” he says, because events get placed in the realm of the logical. ”Believing in fate, or even conspiracy, can sometimes be more comforting than facing the fact that sometimes things just happen.”…We are far too taken…with superfluous facts and findings that have no bearing on the statistics of coincidence. After our initial surprise…the real yardstick for measuring probability is ”How surprised should we be?” How surprising is it, to use this example, that two 70-year-old men in the same town should die within two hours of each other? Certainly not common, but not unimaginable. But the fact that they were brothers would seem to make the odds more astronomical. This, however, is a superfluous fact. What is significant in their case is that two older men were riding bicycles along a busy highway in a snowstorm, which greatly increases the probability that they would be hit by trucks…Statisticians …emphasize that when something striking happens, it only incidentally happens to us. When the numbers are large enough, and the distracting details are removed, the chance of anything is fairly high. Imagine a meadow, he says, and then imagine placing your finger on a blade of grass. The chance of choosing exactly that blade of grass would be one in a million or even higher, but because it is a certainty that you will choose a blade of grass, the odds of one particular one being chosen are no more or less than the one to either side…One relatively simple example of this is ”the birthday problem.” There are as many as 366 days in a year (accounting for leap years), and so you would have to assemble 367 people in a room to absolutely guarantee that two of them have the same birthday. But how many people would you need in that room to guarantee a 50 percent chance of at least one birthday match? Intuitively, you assume that the answer should be a relatively large number. And in fact, most people’s first guess is 183, half of 366. But the actual answer is 23. In Paulos’s book, he explains the math this way: ”[T]he number of ways in which five dates can be chosen (allowing for repetitions) is (365 x 365 x 365 x 365 x 365). Of all these 3655 ways, however, only (365 x 364 x 363 x 362 x 361) are such that no two of the dates are the same; any of the 365 days can be chosen first, any of the remaining 364 can be chosen second and so on. Thus, by dividing this latter product (365 x 364 x 363 x 362 x 361) by 3655, we get the probability that five persons chosen at random will have no birthday in common. Now, if we subtract this probability from 1 (or from 100 percent if we’re dealing with percentages), we get the complementary probability that at least two of the five people do have a birthday in common. A similar calculation using 23 rather than 5 yields 1/2, or 50 percent, as the probability that at least 2 of 23 people will have a common birthday.” Got that?”
The Odds of That, New York Times
Lisa Belkin

The great traders “get” all this. Do you?