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.)
This ought to be interesting. Let's call the two halves L and R and number the cards in order 1-26. The first riffle will give one of two results
L1 R1 L2 R2 ...
R1 L1 R2 L2 ...
So give any card, we'll know the second card 50% of the time
The second riffle (no relabelling of cards) gives one of 4 results
L1 L14 R1 R14 ...
L14 L1 R14 R1 ...
R1 R14 L1 L14 ...
R14 R1 L14 L1 ...
Which is now very interesting. Given a card like L1, we know the next card is L14 (2), R14 (1), or R15 (1), which if we're gambling, still means we'll get it right 50% of the time. Needless to say, I believe this problem deserves the difficulty rating it has...but I look forward to seeing this community tackle it.
I've always wanted to know this ...
