Call a biased coin a p-coin if it comes up heads with probability p and tails with probability 1-p. We say that a p-coin simulates a q-coin if by flipping a p-coin repeatedly (some fixed finite number of times) one can simulate the behavior of a q-coin.

For example, a fair coin can be used to simulate a 3/4-coin by using two flips and defining a pseudo-head to be any two-flip sequence with at least one real head. The chance of a pseudo-head coming up is 3/4, so we have simulated a 3/4-coin.

1. Find a rational value p such that a p-coin can simulate both a 1/2-coin and a 1/3-coin, or prove that no such value exists.

2. Find an irrational value p such that a p-coin can simulate both a 1/2-coin and a 1/3-coin, or prove that no such value exists.

You can always simulate a ½-coin using a p-coin: toss it two times in a row, and if you get Heads-Tails, call it "Heads"; Tails-Heads, call it "Tails", and in other cases, toss again.

Thus, the first problem is solved by taking a 1/3-coin, and the second one with a 1/√3-coin -- toss it twice in a row, and if you get two heads, call it "Heads", and "Tails" otherwise.

Posted by Old Original Oskar! on 2005-02-18 10:51:17