Conundrum: If I roll a die of unknown sides, can I have any degree of confidence about the size of the die based only on that single outcome (aside from the tautology that it must have at least the size of the roll)?
Let's say I roll a 6. To what degree (if at all) am I justified in saying that the die is more likely to be 6-sided than 12-sided?
On one hand, rolling a 6 on a 12-sided die isn't anything remarkable; any single roll is unremarkable in isolation. On the other hand, rolling 6 on a 12-sided die is less likely than rolling 6 on a 6-sided die. Which of these cases are "more important" to this question?
For example: can we invoke Bayes factor?
H1 = 6-sided die
H2 = 12-sided die
O = we rolled a 6
K = P(O | H1) / P(O | H2)
P(O | H1) = 1/6 and P(O | H2) = 1/12
K = 12/6 = 2
Which shows non-trivial confidence that the die is 6-sided rather than 12-sided. Am I in the right for thinking like this?