[dubya]

Sort of, depending on what you mean by random. The article seems to imply the uniform distribution on all 52! orders, which 7 shuffles won't get you. For example, you can't reverse a deck with 7 shuffles. There are bets you can put on a deck that's been shuffled 7 times that pay better than the same bet on a uniformly selected order.

[omaranto]

Yeah, that phrase "truly random" in the wikipedia page is misleading. At least they do include in the references a link to Bayer and Diaconis' paper. What the paper says is that (for a certain mathematical model of the riffle shuffle, which experimentally seems to match actual results produced by human riffle shufflers) if you measure the total variation distance d between the probability distribution of the order you get after k shuffles and the uniform distribution on all 52! orders, you get d=1 for k=1,2,3,4 (i.e., maximally far in total variation distance from uniform), d=0.924 for k=5, and d drops off exponentially after that:

  k     5     6     7     8     9    10
  d 0.924 0.624 0.334 0.176 0.085 0.043

[bigbird]

The total variation distance is an upper bound for the absolute difference in probability over all subsets of permutations.

In particular, we can consider the set:

   U = {all permuations that have never been seen by a human}

The counting arguments in the article lead us to conclude that the uniform probability of U is very close to 1, i.e. almost all permuations have never been obtained by a human shuffle. The Bayer-Diaconis result implies that for certain types of shuffles the probability of ending up in U is at least

   1 - (the total variation distance above)

So for 8 shuffles, we have a probablity of at least 82.4% of landing in U. This is considerably weaker than "every shuffle is unique".