Guessing the color of the card
Here’s a probability problem that I ran into while discussing some card game with my friends. It has a very natural formulation, but I have never encountered it before. And I still don’t have a solution.
There is a deck of N cards. N is even, N/2 cards are red and N/2 cards are black. The player turns the cards over one by one and tries to guess the color. All cards in the deck must be played.
What is the probability of at least K correct guesses?
Note that this is not equivalent to coin tosses (binomial distribution). The probabilities of guessing the color of the 1st card, 2nd card, etc. are not equal. The player is smart, the player has perfect memory, the player may have a strategy.
I am also interested in the expected number of correct guesses for the given distribution of cards that a player equipped with the best strategy can make. E.g. the last T cards are of the same color, but this fact is not known to the player. How many correct guesses should we expect?