*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.