Let: E = Alexander the Great actually existed C = A coin is minted with the face of Alexander the Great ~A = not A
You want to know P(E|C), which by Bayes is
P(C|E)P(E)/P(C)
which by partition of unity is
P(C|E)P(E) / (P(C|E)P(E) + P(C|~E)P(~E))
So you need only to assess 1 (existed, had a coin) and 3 (not existed, had a coin), where P(E) is your prior probability that Alexander existed. There’s no need to do complicated categorical reasoning.
I think that P(C|E) is pretty close to 1, I would suspect that any great emperor worth his salt would have minted a coin with his head on it. So the crucial point becomes P(C|~E), as it should be: what is the probability that a coin would be minted with a mythological figurehead? That is what is to be analyzed and that is what will drive the probability of his existence. Let’s say that we know that it’s definitely possible that Alexander the Great existed, and it’s also definitely possible that he did not exist, but we know nothing else (that is, coins are the first evidence we want to introduce). This would put P(E) = P(~E) = 1⁄2.
Leaving P(C|~E) unknown, we get:
All this is coherent with common sense: if the ancients were happily minting mythic figures all the times (x close to 1), this means that P(E|C) = .5 = P(E), and coins cannot be used as an argument for Alexander’s existence, while on the other hand if almost never happened (x close to 0), then a coin becomes a sure sign of his historicity, with intermediate values (say x = .3) increasing the odds by a small or a great percentage. Notice one thing of my model: I’ve assumed that P(C|E) is close to 1, and this means that a coin can only increase the probability of Alexander existing, however slightly, but never decrease it. This wouldn’t have been true if there was a possibility that a great emperor did not had a coin minted with his face, which would be your case n° 2.
Let:
E = Alexander the Great actually existed
C = A coin is minted with the face of Alexander the Great
~A = not A
You want to know P(E|C), which by Bayes is
P(C|E)P(E)/P(C)
which by partition of unity is
P(C|E)P(E) / (P(C|E)P(E) + P(C|~E)P(~E))
So you need only to assess 1 (existed, had a coin) and 3 (not existed, had a coin), where P(E) is your prior probability that Alexander existed. There’s no need to do complicated categorical reasoning.
I think that P(C|E) is pretty close to 1, I would suspect that any great emperor worth his salt would have minted a coin with his head on it.
So the crucial point becomes P(C|~E), as it should be: what is the probability that a coin would be minted with a mythological figurehead? That is what is to be analyzed and that is what will drive the probability of his existence.
Let’s say that we know that it’s definitely possible that Alexander the Great existed, and it’s also definitely possible that he did not exist, but we know nothing else (that is, coins are the first evidence we want to introduce). This would put P(E) = P(~E) = 1⁄2. Leaving P(C|~E) unknown, we get:
P(E|C) = P(C|E)P(E) / (P(C|E)P(E) + P(C|~E)P(~E)) =>
P(E|C) = .5 / .5 + x .5 =>
P(E|C) = 1 / (1+ x)
All this is coherent with common sense: if the ancients were happily minting mythic figures all the times (x close to 1), this means that P(E|C) = .5 = P(E), and coins cannot be used as an argument for Alexander’s existence, while on the other hand if almost never happened (x close to 0), then a coin becomes a sure sign of his historicity, with intermediate values (say x = .3) increasing the odds by a small or a great percentage.
Notice one thing of my model: I’ve assumed that P(C|E) is close to 1, and this means that a coin can only increase the probability of Alexander existing, however slightly, but never decrease it. This wouldn’t have been true if there was a possibility that a great emperor did not had a coin minted with his face, which would be your case n° 2.