Hari_Seldon:
It has been shown that 7 random riffle shuffles of a 52 card deck will leave it within 1% of a perfectly random distribution. What is a random riffle shuffle? Divide the deck in two at a random place then randomly choose between the two packs and take the top card from the chosen pack. If you google “Trailing the riffle shuffle to its lair” you will get a pdf of the paper by D. Bayer and P. Diaconis that establishes this result. Warning: heavy going, including a detour into the cohomology of commutative algebras in which an animal called the shuffle idempotent is described.
There is a good youtube video with P. Diaconis discussing this paper. Numberphile has 6 or so videos with Diaconis discussing coin flips dice and things of that nature.
