The Better Angels of Our Nature: Why Vio - By Steven Pinker Page 0,124

to fight everyone, largely because their far-flung empires make almost everyone their neighbors. Certain cultures, especially those with a militant ideology, are particularly prone to go to war.

But Richardson’s most enduring discoveries are about the statistical patterning of wars. Three of his generalizations are robust, profound, and underappreciated. To understand them, we must first take a small detour into a paradox of probability.

Suppose you live in a place that has a constant chance of being struck by lightning at any time throughout the year. Suppose that the strikes are random: every day the chance of a strike is the same, and the rate works out to one strike a month. Your house is hit by lightning today, Monday. What is the most likely day for the next bolt to strike your house?

The answer is “tomorrow,” Tuesday. That probability, to be sure, is not very high; let’s approximate it at 0.03 (about once a month). Now think about the chance that the next strike will be the day after tomorrow, Wednesday. For that to happen, two things have to take place. First lightning has to strike on Wednesday, a probability of 0.03. Second, lightning can’t have struck on Tuesday, or else Tuesday would have been the day of the next strike, not Wednesday. To calculate that probability, you have to multiply the chance that lightning will not strike on Tuesday (0.97, or 1 minus 0.03) by the chance that lightning will strike on Wednesday (0.03), which is 0.0291, a bit lower than Tuesday’s chances. What about Thursday? For that to be the day, lightning can’t have struck on Tuesday (0.97) or on Wednesday either (0.97 again) but it must strike on Thursday, so the chances are 0.97 × 0.97 × 0.03, which is 0.0282. What about Friday? It’s 0.97 × 0.97 × 0.97 × 0.03, or 0.274. With each day, the odds go down (0.0300 . . . 0.0291 . . . 0.0282 . . . 0.0274), because for a given day to be the next day that lightning strikes, all the previous days have to have been strike-free, and the more of these days there are, the lower the chances are that the streak will continue. To be exact, the probability goes down exponentially, accelerating at an accelerating rate. The chance that the next strike will be thirty days from today is 0.9729 × 0.03, barely more than 1 percent.

Almost no one gets this right. I gave the question to a hundred Internet users, with the word next italicized so they couldn’t miss it. Sixty-seven picked the option “every day has the same chance.” But that answer, though intuitively compelling, is wrong. If every day were equally likely to be the next one, then a day a thousand years from now would be just as likely as a day a month from now. That would mean that the house would be just as likely to go a thousand years without a strike as to suffer one next month. Of the remaining respondents, nineteen thought that the most likely day was a month from today. Only five of the hundred correctly guessed “tomorrow.”

Lightning strikes are an example of what statisticians call a Poisson process (pronounced pwah-sonh), named after the 19th-century mathematician and physicist Siméon-Denis Poisson. In a Poisson process, events occur continuously, randomly, and independently of one another. Every instant the lord of the sky, Jupiter, rolls the dice, and if they land snake eyes he hurls a thunderbolt. The next instant he rolls them again, with no memory of what happened the moment before. For reasons we have just seen, in a Poisson process the intervals between events are distributed exponentially: there are lots of short intervals and fewer and fewer of them as they get longer and longer. That implies that events that occur at random will seem to come in clusters, because it would take a nonrandom process to space them out.

The human mind has great difficulty appreciating this law of probability. When I was a graduate student, I worked in an auditory perception lab. In one experiment listeners had to press a key as quickly as possible every time they heard a beep. The beeps were timed at random, that is, according to a Poisson process. The listeners, graduate students themselves, knew this, but as soon as the experiment began they would run out of the booth and say, “Your random event generator is broken. The beeps are coming in bursts. They sound like this: “beepbeepbeepbeepbeep .