Unfair Coin Flip

[pocaracas]

Hammy, the keyword was "simulate".

[Brian37]

The solution is to flip livestock, not coins.

[GUBU]

drop coin on checkerboard

whatever color it most covers, red or black, corresponds to heads or tails

Haha, points for thinking outside the box. Let's add the additional constraint that you can only use the coin and the action of flipping the coin to get a result.

Beat up the person using the unfair coin until he agrees to use a fair one (which you then substitute for a double headed coin. Shhhhhh).

Hey, it's my birthday, I'm drunkish.

[Edwardo Piet]

Hammy, the keyword was "simulate".

Simulate by still using the same normal non-simulated coin lol.

Simulate a flip... or in other words... don't actually flip the coin normally.

So basically... it was a trick question

---

Each flip isn't the same chance though. There's the unfairness, which you can use to your advantage in the method which eliminates it.

That's not what I mean.

Say it's 60% chance to be heads and 40% chance to be tails. Each flip it's that exact same chance. 60% heads, and 40% tails. It doesn't matter how many times you flip it. And redefining and equivocating by calling a bunch of flips a "throw" won't change the reality.

[BrianSoddingBoru4]

I decline to offer a solution. I'm uncomfortable with change. 

Boru

[Tiberius]

Hammy, the keyword was "simulate".

Simulate by still using the same normal non-simulated coin lol.

Simulate a flip... or in other words... don't actually flip the coin normally.

So basically... it was a trick question

---

Each flip isn't the same chance though. There's the unfairness, which you can use to your advantage in the method which eliminates it.

That's not what I mean.

Say it's 60% chance to be heads and 40% chance to be tails. Each flip it's that exact same chance. 60% heads, and 40% tails. It doesn't matter how many times you flip it. And redefining and equivocating by calling a bunch of flips a "throw" won't change the reality.

The coin is flipped normally. In order to simulate a fair flip with an unfair coin, you need to flip it multiple times. I think we established that pretty quickly. It's not a trick question. You use what you have to get an outcome that matches a fair flip.

Your example of it being 60% / 40% every time is true, however when you group together multiple flips the percentages change. That's the key to the simulation, finding two different sequences of flips that have the same probability of occurring.

[vorlon13]

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

---

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.

---

I think my idea is better since it has no math.

[Edwardo Piet]

Your example of it being 60% / 40% every time is true, however when you group together multiple flips the percentages change. That's the key to the simulation, finding two different sequences of flips that have the same probability of occurring.

How would the percentage change in a relevant way if the ratio is the same?

---

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

---

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.

---

It seems more to me like a math problem with the heads and tails thing only used to make it more palatable to the layman. As grouping together multiple flips and calling that 'simulating a flip' seems like cheating to me. Cheating insofar as it's equivocating and what I meant by the trick question thing.

I am very impressed by the math. Not impressed by the redefinition. I can't math well anyhow.

EDIT: Wait how is disregarding HH and TT results allowed?

lol

[Autumnlicious]

Easy - have someone flip the coin and record the results for each run.

Parallelize this over several thousand people after securing IRB approval.

Aggregate the results, apply statistical analysis and compare against an ideal distribution and note if it fits within acceptable tolerances.

Write a paper because you've done everything but for an academic study.

[brewer]

Only supply enough rotation energy to the coin so that it turns/flips over. It goes from flat to edge to flat. The stationary coin at heads up always flips to tails and vice versa. This satisfies the flip requirement and the 50% requirement.