[bigpooch]

This is related to a recent post on SMP.

The "proposition"
----------------

You pay $7 for the privilege of playing this game with an
ordinary deck of cards: 26 red cards and 26 black. You
simply guess red or black through the entire deck one card
at a time and get paid $1 per correct guess, but pay out $1
for each wrong guess. The only information you have are
the cards that have gone by. Obviously, if your first pick
is correct, you'll get at least 27 right (or you shouldn't
lose more than $5 in this case). The deck is randomly
shuffled for each game.

1) What is the optimal strategy? (not hard)

2) What are the chances you'll only guess 26 right, i.e.,
lose $7 (playing optimally)?

3) What is the EV/value of this game?

4) Let n be the number of cards for each color (n=26 in the
original game). What is a good approximation for the
number of correct guesses above n/2 for large n? For which
values of n is the $7 fee too high for the game to be +EV?

For the sake of those trying to solve this, please post
solutions in white.