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# Biasing a fair coin

## Question

You have a coin which is known to be fair: p(H) = p(T) = 0.5 exactly.
Devise a method for simulating a biased coin where
p(H) = 1/π exactly.

Edit:
A gold star awaits a method that requires on average only two flips of the fair coin to produce an arbitrary bias.

## Recommended Posts

• 1

Aiming at gold star...

To smulate a Head with probability P.
Write the desired Probability in binary to n digits

Flip the coin.
if it matches the ith digit (H=1, T=0), flip again.

If it doesn't, announce the  opposite of the result.

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• 1

I feel like such a buffoon!  (I could sum this series in high school, but not now...Please nobody needle me :-( )

Expected number of flips in one evaluation

1/2 of the evaluations stop with 1 flip
1/4 of the evaluations stop with 2 flips
1/8 of the evaluations stop with 3 flips
etc.

Sum of i/(2^i) is 2

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Bingo.

Can you determine the expected number of flips needed for p(H) = 1/π ?

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