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An original math problem
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. |
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