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.*)

Using monte carlo simulation of 1M trials...

When tracking the position of a single card, it's distribution throughout the shuffled deck becomes almost uniform at 10 shuffles. The way the histograms appear in my simulation shows that even at 6 and 7 shuffles, there are large regions in the deck where one would expect to find the card. At 8 shuffles and beyond, it switches and the entire distribution is roughly uniform at a little less than 2% (avg 1.92% = 1/52) with only small regions of slightly lower probability. At 12 shuffles, the max of the distribution is below 1.97%

The question however was about predicting the next card in the deck. Here, even at 8 shuffles there are still large spikes in the likelihood of certain cards following a selected card (on the order of 10-15%) Unfortunately, these spikes don't seem to go away. Even at 12-14 shuffles I see spikes on the order of 5%.