How many times must a pack of perfectly ordered playing cards be shuffled until its order is close to random? That depends on what shuffling method you use. Professor Persi Diaconis of Stanford University, a professional magician who changed career and became a mathematician and statistician, made a thorough study of shuffling. He found that, if you use the riffle shuffle (the usual way card players shuffle cards: cut off about half the pack and riffle the two packets together), it takes just 7 times! If you use the overhand shuffle (another popular method) it takes 2500 times! He also studied some 'academic' shuffles that nobody uses in practice.

References:

Diaconis, Persi (1988). Group representations in probability and statistics. Hayward, California: Institute of Mathematics.

Bayer, Dave and Diaconi, Persi (1992) Trailing the dovetail shuffle to its lair. Annals of Probability, 2: 294-313.