Suppose we have a coin that "tries" to be fair. To be more specific, if we flip the coin n times, and have X heads, then the probability of getting a heads in the (n+1)st toss is 1-(X/n).

[The first time we flip the coin, it truly is fair, with p=1/2.]

A short example is in order. Each row here represents the i-th coin toss, with associated probability of heads and its outcome:

p=1/2: H

p=0: T

p=1/2: H

p=1/3: T

p=1/2: T

p=3/5: T

It certainly would seem that the expected value of X should be n/2. Is this the case? If so (and even if not), then think of this coin as following a sort of altered binomial distribution.

How does its variance compare to that of a binomial [=np(1-p)]?

Does the coin that tries to be fair become unfair in the process? Or does it quicken the convergence to fairness?

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