User blog comment:DontDrinkH20/Expanded Googological Hierarchy/@comment-36543718-20180815185849/@comment-30754445-20180819114000

Randomness is actually a pretty complicated topic.

Technically, it doesn't really matter what process you're using to create random numbers. What matters is whether the resulting sequence of numbers is... well... random. And there are very reliable statistical tests to check these things.

You're right, though, that human brains are terrible random number generators. However, a human hand holding an ordinary 6-sided balanced die, would be a pretty good source of such numbers. Not perfect (because no die is perfectly balanced) but pretty good. If you roll a die 6000 times, you will get each number about 1000 times.

You can do even better, by using some chaotic source of noise. The site random.org, for example, generates their random numbers by measuring fluctuations in the earth's atmosphere. These aren't quantum effects, but the resulting numbers pass every randomness test they've been subjected to (and I'm talking about huge samples of billions of digits). Moreover, you cannot predict these numbers in the long run, even in principle, because its a chaotic system.

At any rate, none of this has anything to do with the problem of "generating a random integer". The problem with that idea is, quite simply, that the number of integers is infinite. So there's no way to "pick an integer at random" which won't be biased in some way.

Think of it in this way: The odds of "a random integer" being less than x is exactly zero, regardless of the size of x.

You can, of-course, pick "random integers" using a biased distribution.

Simple example:

(1) Flip a coin.

(2) If it shows up "heads", flip it again. Continue until is shows up "tails", then stop.

(3) Count the number of heads you got. That's your random integer.

Since theoretically you can get an arbitrary number of consecutive heads, this method can give any non-negative integer. But the odds of a particular number being picked drop rapidly (exponentially, in fact), so that's not what people usually mean, when they say "pick an integer at random from the entire pool of integers".

BTW I've actually developed a far more interesting version, where the resulting "random integer" can even be a googological number. In that version, 50% of the numbers were bigger than fε 0 (10). You could see the results of several "rolls" of this "infinity-sided die" here. But even though my method may seem more exciting than the coin-flipping method, it is just a biased. After all, in reality, exactly 100% of the numbers are bigger than fε 0 (10) :-)