A certain salesman, called Mark O.V. delivers his goods to three cities:

** C1, C2, & C3, ** staying only for one day at a time in each of them.

His stay in

** C1** is always followed by going next to

** C2**.
If he delivers in either

** C2 ** or

** C3 ** he is thrice as likely to continue to

** C1, ** than to the other city.

Provide your estimate of the number of working days in each of the cities

within a period of** 12,000 **working days.

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.*)