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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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re(2): Simple solution | Comment 5 of 9 |
(In reply to re: Simple solution by Steve Herman)

"(a) using a 1/p coin, the probablity of head-tails is not equal to the probability of tails-heads, unless p = 2"

If you have a 1/3-coin, the probability of heads-tails is 1/3 * 2/3. The probability of tails-heads is 2/3 * 1/3, and multiplication is commutative, so the probability is 2/9 in either instance.

On the other hand, problem (b) is valid: "Any solution which includes tossing again does not satisfy the the requirement the "fixed finite number of times" requirement."


  Posted by Charlie on 2005-02-19 02:55:00
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