Here is the setup. Shuffle a deck of 52 cards so their order is random (i.e., determined by a uniformly distributed random variable). Now flip through the cards and find the maximum number of consecutive cards with the same suit. My question is: What is the expected value?
I don't know very much statistics, but I can't seem to find a straightforward way of getting to the answer because you're taking a max over a set of values. Intuitively, it seems like the question may be easier to answer if you consider the deck only up to cyclic order (so you're allowed to get to the top and keep counting from the bottom) because then there are no "preferred" cards. I'd be happy with an answer either way.
Preliminary testing with a deck of cards in my office indicates the answer is probably between 3 and 4.
My DP Python program gives 149116192125870533506434119 / 44703948137907327366031200 exactly, which is approximately 3.335638088740207.