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If you're given a single die to cast, and you have no idea how many sides are on it, you can roll the die... and get a 7. What is the expected value of the next cast, and what is the number of sides on the die?
Without any additional information, you should probably answer "7" and "13", I would think... it's better than saying any other single number. With a few more data points, though, you can give answers that are more likely to be correct.
I dunno. Certainly "7" is the best answer to the first question. The die might be crooked, favoring the side that says 7, and you have no reason to guess any other number. So 7 it is.
The answer to the second question depends on whether you were told that the faces of the die contain numbers from 1 to N, where N is the number of faces (like a normal die) or not. If so, then you can assume the die has at least 7 faces, but there is no way to tell how many. "7" is not more likely to come up on a 13-faced die than a 9-faced die. In fact, 7 is most likely to come up on a 7 faced die (1/7th of the time). Therefore the right guess is "7". If you cannot assume that the numbers 1-6 exist, then all you know is that the die has at least one face. It could have infinitely many faces, but one is all you have information about. For the same reasons, "1" is the correct guess here.
Of course, I am assuming that you were unable to cop a feel of the die when you rolled it. An infinite-faced die will feel somewhat different in your hand than a one-faced die.
barfo
