Not only is there no such thing as a “fair” coin, there are no “fair” dice, either. Or “fair” anything. Which I shall prove to you.
From Khan Academy, a popular, and wrong, view:
What does it mean to say you are flipping a fair coin?
Fair means that the coin has a 50-50% chance of getting HEADS or TAILS. They have that because in some other situations, there are “unfair” coins that have more chance of getting one result or another.
If the evidence we have is “Here is a coin with two sides, H and T, that when flipped must come up one side or the other” the probability of H given this is 1/2. We did not have to add “fair” to this. What would adding “fair” do? Make it even more 1/2? The idea does not make sense.
For comedians who ask about coins landing on their sides, please read the evidence again more carefully.
Our coin example is much the same as this: “Here is a device that must be in one of two states, H or T”. The probability of the device being in state H given this evidence is also 1/2. Here there is no way to meaningful add “fair”.
How about evidence like this? “Here’s a unfair quarter, with two sides, etc.” What are the chances, given this evidence, it comes up H? The best we can say is “not 50%.” But that’s adding to the evidence the implied meaning of unfair and the tacit knowledge that something will go wrong, we know not what. Indeed, there are so many possibilities for tacit and implicit premises here it’s difficult to be coherent.
Coins do not have probabilities. Flipped coins do not have probabilities. Nothing has a probability. Probability is only defined by the evidence you assume. Probability is all in your mind.
Ask my old advisor Persi Diaconis to flip a quarter. What is the chance it comes up H? Well, to you, it is 1/2, if you used something like that evidence above.
But to Persi, who has a coin flipping machine, the probability is 1. The coin will always come up H. I have a fuller description in the talk I gave in Phoenix earlier this year.
The machine was built to control the causes the coin comes up one way or the other. Those causes relate to the physical aspects of the coin, the spin, and the force it’s given. If you know the relevant causes, all of them in all aspects, then you know the outcome.
It’s a simple as that. In every problem. Really.
We don’t need “long runs” or any other such nonsense, which speak of infinity, about which nobody knows anything in any empirical way. Probability is not frequentism. In frequentism no probability can ever be known until infinite trials are conducted. Lots of luck with that. Especially in counterfactual trials which cannot be conducted, but in which we can easily see probabilities. Instead, probability is the degree of truth propositions have given assumed evidence. Probability is logic.
Now for the proof there are no “fair” coins.
When you think of “fair” coins you likely have an idea of the symmetry of the coin. It’s round and of roughly even thickness. But that’s not real symmetry. It’s only approximate. An American quarter has unequal mass across the obverse and reverse. The grooves on the edge are not perfectly equal. So this is not a “fair” coin if symmetry is the definition.
But maybe you think you can manufacture such a coin, taking great care the mass is as uniform as you can get it, along all the coin’s aspects. Alas, you will not succeed. And you won’t be able to know, either. Are you perfectly certain the number of quarks, or whatever, are equal in number across every dissect? No, sir, you are not. And cannot be.
Symmetry fails, then, to describe “fair” coins. (Though, of course, if such a thing existed, knowing this would not change the probability we first computed.) But even supposing an absolutely symmetric coin existed, it still has to be flipped. Then what?
The nail that flips it has friction, which varies. The air in which the coin flies has varying densities. The force exerted from other objects during the flip, like the earth, are varying. The spin varies. The power given the coin, and the location it is given on the coin, varies.
Just what is symmetry supposed to mean in all this?
Well, nothing that is coherent. But we do know that if we control all the causes, at least to an important degree, we can get the coin to do what we want.
So what about your coin in the special weird circumstances you create? The probability we began with is a model. Start with that (your “prior”), and then take evidence and see if your model is useful or not. Or make a new model using that evidence.
There are no such things as “fair” coins, and no such thing as probability.
If you’ve read this far, congratulations, because this has much more importance than you might have thought. Believing probabilities exist makes bad science. Or at least limited and incomplete science. Quantum mechanics, anybody?
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