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
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".
