On Thursday, January 31, 2013 3:23:35 AM UTC-5, William Elliot wrote: > On Wed, 30 Jan 2013, email@example.com wrote: > > > On Wednesday, January 30, 2013 12:29:09 AM UTC-8, William Elliot wrote: > > > > > > There is a fair coin with a different integer on each side that you can't > > > > see and you have no clue how these integers were selected. The coin is > > > > flipped and you get to see what comes up. You must guess if that was the > > > > larger of the two numbers or not. Can you do so with probability > 1/2? > > > > > > Well, there was the monty hall problem years ago. Suppose there are 3 > > > keys and you have to choose one. One of the three turns on a car's engine > > > where the other two do not. After you choose a key Monte takes one and > > > tries it in the car and it doesn't turn over the engine. He then asks > > > you if you'd like to change your mind. What should you do? You should > > > change your mind because the first key you chose had a one in three chance > > > of turning on the car where the remaining key has a two in three chance. > > > > > > Soo... > > > > > > Assuming the coin only has positive integers then you should guess > > > the number showing is the smaller integer because there are more > > > integers greater than x than there are less than x and this is true > > > for all positive integers x. > > > > > Recall, you have no clue how the integers were selected much less that > > both are positive or non-negative. > > > > Here's a variant. Instead of integers, the coin has > > a positive rational on each side.
It's even easier if there is a real number on each side. A method to get better than 50-50 odds appeared in a paper many years ago, written by a Stanford professor named Cover (I think). It's very simple, and the proof that it works is a one-liner.
Here's a hint: You want to pick the side you see if the number on it is "large". How can you decide if a number is "large"?
Note: The range of the possible numbers can be (-oo,oo) or any finite interval, e.g. [0,1], and the basic idea is the same.
Also, the way the numbers on the coin are chosen is irrelevant.