karl_lembke (karl_lembke) wrote,

How likely does it have to be?

Robert McHenry writes at Tech Central Station:
I haven't literally been dreamin' about nuclear weapons, but I've been thinkin' about them more than a little bit. The press, the blogosphere, the commentariat - in whatever form you take your medicine - are divided on the great question: How likely is a nuclear attack on the United States by a rogue state or a non-state actor, i.e., terrorists? And I have no ready-made answer to my derivative but highly pertinent question: How likely does it have to be?

Of course, there's another question involved. "How likely does it have to be?" in order to justify what?

This turns in to what probability math calls an "expected value calculation". What is the chance that some event is going to happen, and what is the impact if it does happen?

One example that's often used in probability classes is that of a "fair game". A game is "fair" if a player tends to break even over the long term. That is, the player is likely to win back exactly as much as it costs to play the game. If a "game" involves rolling a six-sided die, and the player wins $6 every time a six is rolled, then a fair price for playing is $1. On average, the player will win every six rolls, and it will cost six dollars to roll the die those six times.

Expected value calculations show up in other places, as well. We sometimes hear news pieces about some deadly car crash that could have been prevented by the addition of a relatively cheap part – usually on the order of $10. These news pieces are usually accompanied by outrage. "Surely," it is asked, "a person's life is worth more than $10?"
The problem with that question is that we're not talking about $10. We're talking about $10, spread over the entire fleet of cars, versus the number of times that part will actually work. If the fleet size is one million cars, it costs $10 million to install that part in every car. If only one accident of the type that part will prevent is likely to occur over the lifetime of the fleet, then the car company has spent $10 million to save one life. We're shocked that a company might decide it's cheaper to settle in a lawsuit, but we shouldn't be surprised.

Another expected value calculation involves the type of asteroid impact that killed the dinosaurs. We don't know the exact chance of such an impact in any given year, but we know it's pretty darn low. The last such impact was some 65 million years ago. It would seem to follow that the yearly chance of such an impact is one in 65 million. This is actually a higher probability than winning the jackpot in one play on quite a few of the big lotteries. People continue to play the lottery, despite the long odds against winning, simply because the impact of winning would be so great.
The odds that a dinosaur-killer asteroid will hit the Earth are remote, but the impact would be – well – earth-shaking. Figure that 99% of the population would be wiped out. Figure that property damage would be in the tens, maybe even hundreds of trillions of dollars. Let's say the impact would be on the order of $200 trillion. A quick calculation shows that it's worth paying $3.1 million per year to develop some sort of asteroid tracking and deflection system.
And if we run the calculation in terms of lives, it's worth expending up to 92 lives per year (say, in industrial accidents) to accomplish this. (Six billion people, times a one-in-65-million chance of a global disaster.)

Now, we return to an intermediate level of schrecklichkeit. What is the impact of a nuclear bomb deployed against the US or one of its allies? We not only have the number of deaths, probably in the tens of thousands, but also property damage and the psychological effect on the world. Given this sort of fallout, how much expense and inconvenience is a fair trade for preventing it?

That's what people need to be discussing.
Tags: math, politics

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