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Riffling Chance (Posted on 2005-08-30) Difficulty: 5 of 5
When shuffling a deck of cards using a riffle shuffle, one divides the deck in two and lets the two halves riffle down to the table, interleaving as they do so. Assume that a person using this shuffle will always divide a deck of 52 cards exactly evenly, and that the riffle will start equally often from the left as from the right.

The expert dealer that I am, when I perform a riffle shuffle the cards from the two halves always interleave perfectly, the cards alternating from the left and right halves of the deck.

How many times must I shuffle the deck before the probability of correctly guessing the next card down in the deck after seeing a card chosen randomly from some place in the deck will be less than 1.97%? (If the cards were perfectly random, the probability of correctly guessing the next card would be 1/51 = 1.96%)

Bonus: What if there were a 10% chance that, as each card falls during the riffle, the card will be covered by another card from the same half, instead of strictly alternating?

(Assume that the person guessing knows the original order of the cards, the number of times the deck has been shuffled, and the probability of the cards interleaving perfectly.)

No Solution Yet Submitted by Sam    
Rating: 4.3333 (6 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re(2): Pascals triangle | Comment 4 of 10 |
(In reply to re: Pascals triangle by Leming)

Maybe I read to much into the wording, but I see a difference between having the riffle start from either side with equal probability versus starting the riffle from both sides equally often.  As it is worded, I would take it that this puzzle is only defined under an even number of riffles, with exactly half of these (though the order is not important) left first, the others right first.  This will further duplicate situations and increase the required rifflings.

Further, one must be carefule to realize that the chances of correctly guessing a following card would not be (necessarily) be the same foreach of the individual cards, and it is the average over the entire 52 cards deck which must be less than 1.97%.

As if we needed this to be harder.

  Posted by Cory Taylor on 2005-08-31 15:28:54

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