You don't need unbounded time for a single flip, that's all in your imagination. The worst-case time is unbounded, but you can't achieve the worst case.
It's very easy to achieve if someone hands you a "coin" that is made such that it never lands on tails.
Sorry for still being in the adversarial mindset, but this means that you essentially have to hardcode a maximum number of same-side flips after which you stop trusting the coin.
> You don't need unbounded time for a single flip, that's all in your imagination. The worst-case time is unbounded, but you can't achieve the worst case.
There literally isn't a bound, it can be arbitrarily large. This isn't just in my head, it's a fact. If it's bounded in your mind then what is the bound?
It's not a fact of the real world, at least. "You can't achieve it" is true. A pretty small number of failures and you're looking at a trillion years to make it happen. And if you buffer some flips that number gets even smaller.
> A pretty small number of failures and you're looking at a trillion years to make it happen.
This depends on the bias of the original coin. P(H) can be arbitrarily large, making P(HH) the likeliest possibility even for a trillion years. "This wouldn't happen in the real world" would be a sorry excuse for the deliberate refusal to clearly state the problem assumptions upfront.
IMO, if you really want to pleasantly surprise people, you need to be forthcoming and honest with them at the beginning about all your assumptions. There's really no good excuse to obfuscate the question and then move the goalposts when they (very predictably) fall into your trap.
> This depends on the bias of the original coin. P(H) can be arbitrarily large
> There's really no good excuse to obfuscate the question and then move the goalposts when they (very predictably) fall into your trap.
Interesting. Because I see the guy pulling out the one-in-a-million coin and expecting it to run at a similar speed to be doing a gotcha on purpose, not falling into a trap and having the goalposts moved.
And I think "well if it's a million times less likely to give me a heads, then it takes a million times as many flips, but it's just as reliable" is an answer that preserves the impressiveness and the goalposts.
It's fast relative to the bias. Which seems like plenty to me when the original claim never even said it was fast.
(And if the coin never gives you a heads then I'd say it no longer qualifies as randomly flipping a coin.)
