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Peter Scales

Posts: 192
From: Australia
Registered: 4/3/05
Re: (no subject)
Posted: Aug 15, 2012 11:00 AM
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> If a coin is flipped until there has been at least
> one head and at least one tail and gcd(number of
> heads,number of tails) >1, what is the expected
> number of flips?
> I simulated 1000000 games because I couldn't be
> bothered to think and got a mean of 6.00584, and if
> the answer is 6 there ought to be some elegant proof
> that is eluding me.

Hi Angela,
I'm not sure I fully understand the question, but it seems you need at least 4 flips.
After 4 flips you should have achieved the desired result in 1/4 cases, on average.
Presumably successful sequences will be discontinued.
For those sequences not achieving the desired result, a further 2 flips will achieve the desired result in 6/12 cases, on average.
Again continuing the still unsuccessful sequences, a further 2 flips will achieve the desired result in 11/24 cases, on average.
You could continue this sequence. It may or may not verify your conjecture. I hope this helps.
Regards, Peter Scales.

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