What about the wind? What about the force the hand exerts on the coin in tossing it? What about the shape of the floor and its irregularities? How can we know all these factors will add up precisely in the way required to compensate over time any partial slant coming from any of them?

Is no physical theory required to sustain such a belief?

Of course the fairness of the coin decisively matters but how can it be enough to exclude any other possible bias?

Death Monkey wrote:


I don't understand your question. The "fairness" of the coin is not some independant property it has. The coin is defined to be "fair" if and only if there are no biases. That is what "fair coin" means.

DM

The fairness of the coin refers only to the coin's physical construction. But there can be other biases. Suppose the one who flips the coin has a special way to do it that favors heads, which may happen even if all involved agents are completely unaware of the fact. Imagine that the interaction of the specific way in which the coin is tossed once and once again with the wind direction is what favor heads, all agents being unaware thereof. Imagine...

Why are we entitled to believe that a fairly constructed coin will not encounter systematic bias arising from the environment? On what grounds are we to expect that heads will tend to come up 50% of times?

Is this not plainly a case of application of the principle of indifference justified by the absence of further information? Or have we any physical theory that entitles us to believe that the sum of all environmental factors will tend to favor equal distribution of chances?

Not so. The bias can come from natural events in the environement.

But they could yield 100% heads by virtue of the chain of physical causes that leads to the result. I know we usually overlook this but the question is on what grounds are we justified to do so. Experience is no good candidate for mathematical justification in computing probabilities.

Let me show you how the two envelope paradox is no different from the coin toss case by presenting a strengthened version of the paradox. We could call it 'the strengthened Kraitchik paradox' (from Maurice Kraitchik, the French mathematician who proposed the original paradox for the first time). The following is a physically impossible idealized case but such cases are admissible in mathematical reasoning.

A person P makes, say, one billion choices between envelopes A and B in each pair, and gets $10 in, say, 10 occasions. P opens all chosen envelopes and sees what lies in them. Now P is informed that the choice between all possible distributions has been made by tossing a fair coin (an infinite number of times; this is an idealized thought experiment).

Then P is allowed to switch the envelopes in those cases in which the chosen envelope contains $10.

P reasons this way: if I never switch, I’ll get $10·10 = $100 from the envelopes with $10. But if I always switch, the probabilities are that I will get $20 half the times (which already amounts to $100) and $5 the other half (which sums up to %25); in total %125 instead of the $100 that I would get if I always kept the chosen envelope.

The problem is that P could have reasoned similarly if he had chosen the other envelopes in those ten cases. So the paradox arises again. But now the distribution of money into the envelopes depends on the toss of a fair coin.