Paulos: How to Calculate Chances of Doomsday

May 5 -- Is the sky falling? And if so, when? Even when they're baseless, constant reports about nuclear weapons proliferation, pandemic diseases and environmental catastrophes revive these perennial human questions and contribute to a feeling of unease.

So too did the recent passing of an asteroid almost 100 feet in diameter within 30,000 miles of the Earth. Such news stories make a recent abstract philosophical argument a bit more real.

Developed by a number of people including Oxford philosopher Nick Bostrom and Princeton physicist J. Richard Gott, the Doomsday Argument (at least one version of it) goes roughly like this.

There is a large lottery machine in front of you, and you're told that in it are consecutively numbered balls, either 10 of them or 10,000 of them. The machine is opaque so you can't tell how many balls are in it, but you're fairly certain that there are a lot of them. In fact, you initially estimate the probability of there being 10,000 balls in the machine to be about 95 percent , of there being only 10 balls in it to be about 5 percent.

Now the machine rolls, you open a little door on its side, and a randomly selected ball rolls out. You see that it is ball number 8 and you place it back into the lottery machine. Do you still think there is only a 5 percent chance that there are 10 balls in the machine?

Given how low a number 8 is, it seems reasonable to think that the chances of there being only 10 balls in the machine are much higher than your original estimate of 5 percent. Given the assumptions of the problem, in fact, we can use a bit of mathematics called Bayes' theorem to conclude that your estimate of the probability of 10 balls being in the machine should be revised upward from 5 percent to 98 percent. Likewise, your estimate of the probability of 10,000 balls being in it should be revised downward from 95 percent to 2 percent.