> If you shuffle a deck thoroughly, then you have produced something that the
> world has never seen before, and if left only to chance, will never see again. If
> a trillion people shuffled a trillion decks of cards each, a trillion times per
> second, and had been doing so since the universe began 13 billion years ago,
> the chance that one of them would have produced the same order of cards you
> did would be about one in a trillion trillion.
Um, no. Let me explain:
First, 52! is a little less than (10^68), so there are that many different orderings of the cards.
A trillion is 10^12. If a trillion people shuffled a trillion decks of cards each a trillion times a second, that would mean that there would be
(10^12) X (10^12) X (10^12) = (10^36)
different orderings of the cards per second.
The number of seconds in 13 billion years is
1.3 X (10^10) X 3.65 X (10^2) X 2.4 X (10^1) X 3.6 X (10^3)
so there are a little more than (10^17) seconds in 13 billion years.
This means that the total number of different orderings these people would have produced is (10^53).
You seem to think that it would be necessary to produce about (10^68) different orderings to expect to see even one case where two of those orderings match. That’s not true. It’s only necessary to produce about (10^34) different ordering to expect to see about one case where two of those orderings match.
This is an example of what’s called the Birthday problem:
Suppose you want to pick a group of people just barely large enough that you can expect one pair of those people to have the same birthday. You don’t need 365 people. You only need 23 people. Approximately speaking, if there are n different values that something can take, you only need a group of about the square root of n different things chosen randomly for there to be about one match among them. (Well, it’s not exactly the square root, but since we’re only dealing with numbers to the nearest power of 10, that’s close enough.) So in the situation you describe, there would be an enormous number of matching pairs of orderings.
Someone please check my arithmetic, since it’s easy to mess up in cases like this.