Timothy and Urban play a game with two dice. But they do not use the numbers. Some of the faces are painted red and the others blue. Each player throws the dice in turn. Timothy wins when the two top faces are the same color. Urban wins when the colors are different. Their chances are even.

The first die has 5 red faces and 1 blue face. How many red and how many blue are there on the second die?

If you have a collection of coins, you can use the entire collection as one coin by flipping them all, and announcing heads if an even number of heads turn up, tails otherwise.

RESULT: The entire collection of coins simulates a fair coin precisely if any one of the coins is fair.

PROOF: Given a coin, let the BIAS be the probability of heads minus the probability of tails. This is a number between -1 and 1 inclusive.

The BIAS is zero if the coin is fare. The BIAS of the simulated coin is equal to the products of the BIASes of the coins in the collection.

This can be seen as a negative result: you cannot create a fair coin this way without having a fair coin to start with. However since all the BIASes are numbers in the range [-1,1], the more coins you use, the less Bias there will be.

Posted by Luke on 2003-04-01 07:35:32