Friday, November 14, 2008

Unholy brain-teaser

To pass time in Heaven (there’s a lot of time around), Gabriel says to God: “Our Father, who art right here, might Thou deign to think of a number, any positive whole number, entirely at random?” God, being supremely benevolent, and also a bit bored, agrees. Gabriel says: “Now, if it doth please Thee, my Lord and shepherd, write the number down (or possibly carve it on stone tablet).” Again, God does so.

“And finally, O great and infinite creator of all that is good and true, Thy humble servant beseecheth Thee to hang around for a little while, for I am shortly going to ask Thee to think of a second, different number, any positive whole number, entirely at random. But don’t think of it just yet!” God murmurs: “THIS HAD BETTER BE GOING SOMEWHERE.”

“But it is, most svelte and magnificent one. I am going to bamboozle a newcomer.” Gabriel then turns to you (you’re in Heaven too, following a freak thumb-twiddling accident and perhaps, for all I know, a clerical error by St Peter) and says:

“Right. I’m taking bets on which of these two numbers will be larger. What do you say?”

(a) The first number will probably be larger than the second; I’ll bet a month’s supply of ambrosia.
(b) The second number will probably be larger than the first; I’ll bet a month’s supply of ambrosia.
(c) The first number will almost certainly be larger than the second; I’ll bet my immortal soul.
(d) The second number will almost certainly be larger than the first; I’ll bet my immortal soul.
(e) They’re both random! Either number is equally likely to be larger; I’m not betting on what’s basically a coin-toss.

So, assuming that God has followed Gabriel’s directions perfectly (He is perfect, after all) – and assuming that gambling isn’t sinful – what do you say? Why?


Anon said...


Whatever number God thought of first, there are infinitely more numbers greater in value for him to choose from than there are numbers smaller in value.

Tom Freeman said...

When I first came up with this puzzle, that’s what I thought.

If you ask people to think of “any positive whole number, entirely at random”, they will generally not do that. We’re biased towards the (low) sorts of numbers we’re familiar with. But God, with an infinite mind not constrained by human perspective, would pick truly randomly from the infinity of positive whole numbers.

Say that his first choice is huge (by human standards): a trillion-digit number, n. (If he carves stone tablet at the rate of 100 digits per second, it will take 317 years to get n down. He’s probably faster than that, what with the omnipotence.)

Now, given this first number, any subsequent different number can be either smaller (n–1 possibilities) or larger (infinite possibilities). It is overwhelmingly likely that the second random number will fall into the infinitely bigger latter group. So, whatever the first number is – you really don’t need to know – the second will almost certainly be larger: (d).

But this reasoning is wrong. Why?

Eunoia said...

Statistically speaking, I would expect the first number to be half way to infinity. So that's also infinity. So it takes forever for Him to pronounce its digits. Therefore they never get around to the second part of the bet ;-)

Tom Freeman said...

This is what I reckon.

Imagine if God were picking the two numbers simultaneously (but still independently). Surely then it would be fifty-fifty… The argument for (d) depends on one of the numbers being treated as the threshold to judge the other against, and this choice needs a solid, non-arbitrary reason. But ‘being picked first’ won’t do. It might make some intuitive sense to us, but the laws of mathematics don’t care what happens when.

There’s no objective reason to privilege the first number picked, and so you could apply the ‘this random number will almost certainly be larger than that other random number’ logic equally well both ways. Which means you can’t apply it either way.

The paradox comes from the fact that the key assumption behind the picking scenario is false.

Imagine if God were asked to pick 100 whole numbers between 1 and a million at random. You’d expect them to be spread pretty well across the range. It’s possible that they might all be clustered, say, between 1 and a thousand, but this would be extremely unlikely.

Now think what happens if God picks 100 numbers at random from the whole, infinite range. Say the smallest of these is x and the largest y. This range, from x to y, however wide, will cover only an infinitesimal fraction of the available range, making it almost totally unlikely that all the numbers picked would fall within it. But they do. And this result will happen whatever numbers are picked, however huge the range from x to y. The extraordinarily unlikely must happen.

Which means that it’s impossible to pick truly randomly from an unbounded infinite set. Even if you’re God.