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by vibragiel
5667 days ago
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I think this problem has no satisfactory answer. We are asked to operate with quantifiable information —probability of special and normal coins coming up heads— and unquantifiable information —my "belief" in a coin being a special coin, which may, or may not, depend on my level of knowledge of several factors, like the plausibility of the existence of special coins, their prevalence, my competence identifying them... EDIT: ...my friend being or not a usual liar, his sleight-of-hand skills, my level of rational analysis and critical thinking, me being on drugs, me dreaming, me actually experiencing the Matrix... We are trying to quantify the unquantifiable: my "belief" in something, a psychological phenomenon which depends on millions of rational and irrational factors. |
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My reasoning, for what it's worth, goes as follows.
It's surely impossible to make a coin that literally always comes up heads no matter how you toss it. (Imagine that you contrive to send it vertically into the air with no spin at all. It will come down the same way as it went up.) So let's charitably interpret my hypothetical friend's claim as something like "if you toss this in a normal sort of way, it'll come up heads at least 99% of the time". That seems like it might be achievable with something that looks like a normal coin. Now, how likely is that given only that (1) it looks like a normal coin and (2) my friend has told me what he has?
Well, now it's time to pluck numbers from the air, in an attempt to quantify my beliefs about biased coins and my friend's trustworthiness and so forth. I suppose I think there's about a 50% chance that it's actually possible to make a coin that behaves as described. If so, it probably isn't particularly hard to figure out how to do so, in which case the makers of magic tricks and practical jokes and suchlike will surely do it and sell the fake coins at a reasonable price. Except that in most countries it's a serious offence to sell fake currency, so really convincing fake coins would have to come from abroad and be shipped in surreptitiously. Let's say a 10% chance, given that such coins are possible and not too expensive to make, that it's commercially worth while for someone to do that.
Now, my friend is purely hypothetical, so who knows what his character and interests might be? Let's suppose he's known to be interested in this sort of thing. Then perhaps he's about equally likely to procure such a coin, if it exists, as to play a joke on me by claiming to have such a coin and just hoping it comes up heads. (Of course if no such coins exist then he'll certainly do the latter.)
So, my prior probability for the coin's being fake in the appropriate way is about 1/20.
Now Pr(coin is fake|heads) / Pr(coin is normal|heads) = Pr(coin is fake) / Pr(coin is normal) times Pr(heads|coin is fake) / Pr(heads | coin is normal). That's 1/20 times 0.99/0.5, or about 1/10.
So, before the coin toss I'd have given roughly 20:1 odds against the coin's being fake; after it, roughly 10:1.
(Note: I have deliberately done the calculations only very roughly. That's usually good enough in practice, and all my prior probabilities are crude estimates anyway.)
Of course, if (e.g.) my friend starts offering suitable bets on the outcome of the next few tosses, that will itself be evidence that the coin really is fake. So if you happen to have such a coin, you won't be able to cheat me out of as much money as you might hope :-).