Given that the coin either has two of the same faces or is fair, and it is not more likely to be two-tailed than two-headed...I think that you are asking what the evidence indicates the bias of this one coin is, rather than asking if this one coin is biased. Which, in fact, does apply to the case I was trying to illustrate.
In what sense is a "fair coin" not a coin with a 'bias' of exactly 0.5?
The first throw being heads is evidence for the proposition that the coin is biased towards heads, evidence against the proposition that the coin is biased towards tails, and neutral towards the union of the two propositions.
If showing heads for the first throw was evidence for or against the coin being fair, not showing heads for the first row would have to be evidence against or for.
That we survived is evidence for "MAD reduces risk" and evidence against "MAD increases risk", but is neutral for "MAD changes risk".
The standard view of Mutually Assured Distruction (MAD) is something like:
Occasionally people will reply with an argument like:
This is an anthropic argument, an attempt to handle the bias that comes from a link between outcomes and the number of people who can observe them. Imagine we were trying to figure out whether flipping "heads" was more likely than flipping "tails", but there was a coin demon that killed everyone if "tails" came up. Either we would see "heads" flipped, or we would see nothing at all. We're not able to sample from the "tails: everyone-dies" worlds. Even if the demon responds to tails by killing everyone only 40% of the time, we're still going to over-sample the happy-heads outcome.
Applying the anthropic principle here, however, requires that a failure of MAD really would have killed everyone. While it would have killed billions, and made major parts of the world uninhabitable, still many people would have survived. [1] How much would we have rebuilt? What would be the population now? If the cold war had gone hot and the US and USSR had fallen into wiping each other out, what would 2013 be like? Roughly, we're oversampling the no-nukes outcome by the ratio of our current population to the population there would have been in a yes-nukes outcome, and the less lopsided that ratio is the more evidence that MAD did work after all.
[1] For this wikipedia cites: The global health effects of nuclear war (1982), Long-term worldwide effects of multiple nuclear-weapons detonations (1975). Some looking online also turns up an Accelerating Future blog post. I haven't read them thoroughly, and I don't know much about the research here.
I also posted this on my blog