On Thu, Jun 3, 2010 at 1:27 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Here's the discrete version of the puzzle, where I don't know if the median trick is helpful:
---------- Suppose two numbers are chosen from {1,2,3,...,52} without replacement, independently at random. What is the best real number guess G (in closed form) for the value of the smaller number, if once again best means having least expected absolute error in the long run?
For large values of 52, the answer to the continuous problem should be a good approximation. So I would guess the answer is close to 1 + (52 - (51 * (sqrt(2)/2)) = 16.9. So the answer is probably 17, and if not 17, 16 or 18. The median trick still works. Out of 52 * 51 = 2652 possibilities, The answer is 17 or greater if both cards are 17 or greater. There are 36 * 35 = 1260 ways this can happen, a bit less than 1/2. The answer can be 16 or greater in 37 * 36 = 1332 ways, a bit more than 1/2. So the median value is one of the 72 possibilities where the answer is exactly 16, and the guess that minimizes mean error is 16. Andy