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.)
From the web:
A perfect shuffle is where the deck is divided exactly in half and the two halves are interleaved perfectly; left, right, left, right, ...
If you divide the deck in half and the top cards go into the left hand and the bottom cards to the right, then an
"in shuffle" is where the first card comes from the left and an
"out shuffle" is where the first card comes from the right.
8 perfect out shuffles return the deck to it's original state. It takes more in shuffles to do the same thing.
According to this site, 7 riffle shuffles are sufficient. More shuffles do not affect randomness. This must assume that you're not a card sharp and able to do 8 perfect out shuffles.