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.