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.)

I agree with Bob and how he started tackling this.  The probability that a given card will follow the randomly chosen card is based on the original distribution and Pascals triangle.

With one "riffle" only two cards could be in the next card slot. With two riffles, three cards could be in the next card slot, with one having twice the probability of the others. 

           1

        1    1

     1    2    1

  1    3    3    1

1    4    6    4     1

Two concerns: It looks like a LOT of riffling is required, and the effects of having a finite number of cards will alter the equation.

Posted by Leming on 2005-08-31 00:32:37