GUY RAZ, host: Now, for the past decade, U.S. military officers in Iraq and Afghanistan have struggled to protect troops against surprise insurgent attacks. Roadside bombs, suicide attacks, ambushes, all methods used by insurgents, and difficult for a well-equipped military to defeat. So in the hope of mitigating those attacks, some military strategists are increasingly looking to math, specifically the study of probability.

Neil Johnson is a physicist at the University of Miami, and he's come up with a mathematical formula that he believes can help predict future insurgent attacks. And now, the Pentagon is beginning to put Johnson's ideas to the test.

Dr. NEIL JOHNSON: We all assume it's like some horrible throw of the dice, very much like we assume that a market is a flip of a coin. And we know in times of crisis, that's actually not what happens at all. And it's exactly the same type of thing that we found with the insurgency, that there was a kind of rhyme and reason behind the numbers. They weren't just accelerating, they were accelerating in a particular way.

RAZ: So you looked at insurgent attacks primarily in Afghanistan and Iraq over the past decade. You started to graph them, right, on a chart and you noticed that it was basically a curve. It was a - there was almost an amazing kind of, I guess we could say coincidental, but not coincidental regularity.

JOHNSON: That's right. It's not just a coincidence. It follows exactly the same mathematical pattern, which is called in these other fields a learning curve or progress curve.

RAZ: You based your sort of equation on the Red Queen theory. What's - first of all, what's the Red Queen theory?

JOHNSON: As we know from Lewis Carroll's "Alice in Wonderland" books, and particularly "Through the Looking Glass," the red queen is running on the spot. And unless she keeps running at the same speed, she won't keep up. And, of course, the implication is that everybody else is running at the same speed as well.

And so you get this picture of some kind of neck and neck race, this kind of sporadic pulling ahead of the red queen, which is the insurgency, and the pulling back of the coalition as they realize how to counteract. If you look at that pattern over time, it reproduces exactly this learning curve behavior that I was describing earlier.

RAZ: What is the exact equation?

JOHNSON: It's a very simple one. You can put it into a handheld calculator.

RAZ: OK.

JOHNSON: It says that the time interval between, say, the 100th and the 101st fatal attack is given by the time interval between the first two attacks multiplied by, in this case, by 100 to the power of minus, some negative quantity. And that negative quantity is the escalation rate for the attacks in that particular province.

RAZ: OK. I'm going to trust you that this works. But, say, two attacks happen in the same place, Kandahar province, for example, and they're 100 days apart. So you would then take 100, and you would plug it into your equation, and then the next attack would most likely happen when?

JOHNSON: It would turn out to be about 66 days. It gives you an estimate, admittedly just an estimate, but better than not knowing anything, I would claim.

RAZ: Say, there was no attack after 66 days or, you know, even more, what would that suggest? Would that suggest that maybe there was a problem with the insurgents?

JOHNSON: The learning curve is more than just giving you this kind of sense of how long things are going to take in the future. You can also look at specific moments where that, exactly as you just said, happens. There's some kind of jump, and it's out of what was expected. You can then begin to look behind the scenes. For example, if it was the military, they could say, well, did we do something around that time? Whether we knew it or not, that seemed to have disrupted that learning curve. And if it wasn't them, then it must have been on the red side. And so you've immediately got some kind of new insight that we wouldn't have been able to get otherwise.

RAZ: That's Neil Johnson. He's a physicist at the University of Miami. You can see that mathematical formula he developed to predict insurgent attacks at our website, npr.org. Neil Johnson, thank you so much.

JOHNSON: It's a pleasure. Thank you.

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