Unfair Coin Flip

[Tiberius]

I think Steel got it. I'll wait because I assume he's frantically typing out the full answer.

[Foxaèr]

If it has to be simulated through a computer program, it's hardly feasible for it to be done accurately and upon one's first attempt using one's two hands and a normal coin, correct?

[SteelCurtain]

Not frantically. How dare you.

[Tiberius]

If it has to be simulated through a computer program, there's hardly a way for it to be done accurately and upon one's first attempt using one's two hands and a normal coin, correct?

No need to use a computer program.

[Foxaèr]

Does the coin have to be flipped?

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Or does it only have to be flipped once?

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So long as at least only one flip has to be performed, I say just drop the coin directly onto a flat surface 50 times, then flip it once to the other side and drop it 50 more times onto a flat surface.

[Tiberius]

Ok, well here's the solution that Steel hinted at.

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Let's assume that the coin lands on "heads" with a probability of A, and lands on "tails" with a probability of B. We don't know what A and B are, we suspect they aren't equal.

If you flip the coin twice, there are 4 possibilities with the following probabilities (H = heads, T = tails):

HH = A*A
TT = B*B
HT = A*B
TH = B*A

Now A*B = B*A so the probability of flipping heads and then tails is the same as flipping tails and then heads.

Knowing this, we can ensure that we get a fair result by following this method:

1) Flip the coin twice. If the result is HH or TT, repeat this step.
2) If the result is HT, we say the overall result is "heads".
3) If the result is TH, we say the overall result is "tails".

To show this with numbers, let's say the probability of heads is 70% and the probability of tails is 30%. The results of flipping a coin twice would be:

HH = .7 * .7 = .49
TT = .3 * .3 = 0.09
HT = .7 * .3 = 0.21
TH = .3 * .7 = 0.21

So as you can see, when you disregard HH and TT results completely, you have a perfect 50/50 balance between HT and TH, making the unfair coin produce fair results.

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[SteelCurtain]

Damn I'm smort.

Nods to Poca for the assist.

[Foxaèr]

From my perspective, the theory of chaos would ensure that there is absolutely no way one can get a 50-50 result on the first attempt.

The formula above looks pretty, and I certainly don't understand it, but it seems more like a formula of what could be rather than what absolutely will be.

[Tiberius]

From my perspective, the theory of chaos would ensure that there is absolutely no way one can get a 50-50 result on the first attempt.

The formula above looks pretty, and I certainly don't understand it, but it seems more like a formula of what could be rather than what absolutely will be.

You're guaranteed to get a 50/50 result if you ignore HH or TT flips. The probability of flipping heads and then tails is exactly the same as flipping tails and then heads, regardless of the individual probabilities of heads and tails. This is because it doesn't matter which order you multiple two numbers in, they result in the same value.

[Foxaèr]

But there is still absolutely no way I can grab a coin right now, do precisely one hundred flips, and arrive precisely at 50 heads and 50 tales just from those one hundred flips. That's my point.