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According to the Gilbert-Shannon-Reeds model (which apparently models reality well), one should riffle shuffle seven times to achieve a suitably randomized $52$ card deck.

However, it occurs to me that in bridge as well as many other card games, we don't care about a random order over the entire deck, but only a random order modulo $4$, since everyone is dealt a $13$ card hand (with each person receiving a card in turn).

Given this easier situation, how many riffle shuffles necessary to achieve suitable randomness? I am interested if anyone has any resources for this specific case (and of course I would be very impressed if anyone actually goes through the analysis here).

I understand that suitable randomness is not a well-defined concept - in the Gilbert-Shannon-Reeds model, seven shuffles corresponds to a total variation distance of $0.334$, so I would be looking for the amount of shuffles which gives a similar number.

This question is mostly motivated by a desire to justify my laziness to fellow bridge players.

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  • $\begingroup$ In the James Bond book that features Drax, he cheats at bridge by having little bit of information about the deal, along with good skills in the first place. I suggest you do the full shuffle. Possibly Moonraker, the book may or may not have the same title en.wikipedia.org/wiki/James_Bond#Ian_Fleming_novels $\endgroup$
    – Will Jagy
    Commented May 11, 2015 at 0:34
  • $\begingroup$ Here in the video, the mathematics researcher specifically talks about shuffling in bridge : youtu.be/… and around the 6 minute mark, he talks about the mod 4 behavior. It doesn't exactly answer your question though, you might want to email the professor directly if it interests you. Or send him a link to the question here, maybe he'll answer it for all of us. $\endgroup$
    – DanielV
    Commented May 11, 2015 at 2:02

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