User blog comment:Ikosarakt1/Apocalyptic function/@comment-5529393-20130321020848/@comment-5529393-20130321222303

When the probabilities decay exponentially, the sum of the probabilities do not add up to anything significant. As an example, suppose we are looking for foo numbers, and the probability of a number n being foo is about 1/10^n. Then the expected number of foo numbers is

1/10 + 1/100 + 1/1000 + ... = 1/9.

and the probability of any number being foo is less than the expected number 1/9. So the probability of any number being foo is barely more than the probability that the number 1 is foo. Even worse, the expected number of numbers greater than a googol being foo is

1/10^(googol+1) + 1/10^(googol+2) + ...

= 1/googolplex (1/10 + 1/100 + 1/1000 + ...)

= 1/(9 googolplex)

So the probability that any number greater than a googol is foo is less than 1/(9 googolplex) so you can bet your house it won't happen.

The same principle works for nonapocalyptic numbers. If you add up the probabilities of being nonapocalyptic for all numbers from tritri to infinity, the total number will be on the order of 1/2^tritri. So we can rule it out, the probability is less than the Earth getting destroyed tomorrow. It doesn't matter that the search space is large or even infinite, no number is going to "slip through" since we've accounted for all of them.

For more down to Earth numbers, the probability of a nonapocalyptic number greater than 2^50000 is less than 1 in 1000, and the probability of a nonapocalyptic number greater than 2^10000 is less than 1 in a billion. That's the sum of the probabilities for all numbers up to infinity, not just one number.