# How special is your random shuffle? [duplicate]

I shuffle my standard deck of 52 cards really, really well - a completely random shuffle. This generates some configuration of cards: 9Clubs, 4Spades, KHearts, etc...

What are the odds that this exact same configuration has been generated before? By before - I mean anytime before and anywhere - including all the poker games in all the world and all the magic tricks and all the kids just practicing and all their intermediate shuffles as well.

## marked as duplicate by Rahul, user147263, Claude Leibovici, Jonas Meyer, TrueDefaultNov 5 '14 at 6:31

• Not a super hard or deep question, but I enjoyed thinking about it - made me feel special. – Mark McClure Nov 5 '14 at 4:47
• So, I guess the question is mainly, how many shuffles have ever taken place in the world? A Fermi problem, I suppose. – Rahul Nov 5 '14 at 4:53
• @Rahul Yes, to some extent. I think, though, that a reasonable upper bound (or even an unreasonable one) suffices to answer the question with near 100% certainty. How unreasonable you can be with your upper bound and how near to 100% is the fun part. – Mark McClure Nov 5 '14 at 4:56
• @Rahul - Darn! I did search for such a thing for some time. I guess the closure police will be here soon. :) – Mark McClure Nov 5 '14 at 5:44

Lets just say 10 million poker games are played per day, and lets go back 20 years (I'm guessing before computers the number was neglegable). That's $2\cdot 10^9$ games. The probability of a single predetermined shuffle is $1/52!\approx 1.2\cdot 10^{-63}$. Thus the expected number of times your configuration has been played over the course of 20 years is around $4\cdot10^{-54}$.
On the other hand, $(52*51*50*49*48*47)$ is about 14 billion, which corresponds to asking what the probability is of matching the first 6 cards of your predetermined shuffle, which is much more reasonable to occur.
• Your last statement should be $\frac{1}{\,_{52}P_{12}}$ = $\frac{1}{\binom{52}{12} 12!}$ because you still need to choose the $12$ cards that you're organizing $12!$ different ways. – Axoren Nov 5 '14 at 5:28