**Andrew Robinson wrote:**
"As the number of possible shuffles is so high, an analysis of a sample doesn't seem to be nearly accurate enough."

I don't share your intuition. Perhaps I'm not following the problem. Can you say more about that? And more about why you want exact probabilities? It's a pretty easy experiment to simulate.

I just figure, that as there are 52! outcomes, taking a sample size of even a trillion shuffles does not seem to be a big enough sample size. And if you start taking larger samples, there is too much data to analyse.

So to give further clarification to the problem:

A friend and I were taking about the fact that no properly shuffled deck of cards has (almost certainly) ever been put into that order before, and likely will never be randomly shuffled into that order again.

We then got into what the chances would be that a random - "properly" - shuffled deck would contain certain poker hands. But the numbers started to baffle us.

For example, in looking for a pair, there are 12 possible combinations for any card to make a pair. In this scenario, order matters, i.e., Ace(Hearts) Ace(Clubs) would be an independent outcome from Ace(Clubs) Ace(Hearts). But then this pair can also appear in any 2 consecutive positions along the shuffle. It could be the first and second position, it could be the 30th and 31st positions etc. All of these are seperate outcomes.

Given there are 51 possible positions for a pair to appear, and 13 suits, it would seem to me that the number of pairs are 12 x 51 x 13 = 7956. BUT, in our case, the order of the remaining cards is important, so there must be 50! possibilities. Is that correct? Because (7956 x 50!)/52! leaves a 300% chance of a unique shuffle containing a pair anywhere in the order!

To be honest, I have truly twisted my brain thinking about it!

Any help would be greatly appreciated!