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Biased Coins (Posted on 2005-02-17) Difficulty: 4 of 5
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.

See The Solution Submitted by David Shin    
Rating: 4.0000 (3 votes)

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Some Thoughts re: Simple solution | Comment 2 of 9 |
(In reply to Simple solution by Old Original Oskar!)

The simulation of a 1/2 coin entails (possibly, not probably) infinitely many tosses, so this doesn't fulfill the part about "some fixed finite number of times".
  Posted by Federico Kereki on 2005-02-18 13:27:34

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