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Can we assign probabilities based only on a known number of possibilities? I have two concrete questions below that get at what I'm asking--please answer those if you answer. (I think they are two versions of the same question, but I may be wrong about that.)

Version 1

I find a six-sided die on the ground. I have no information about the die other than the following:

  • It is possible for the die to land on any of the six sides when rolled. (I know this because I have set the die on each side without it rolling over. I have not made any observations about it's shape or balance that would tell me whether or not it is fair.)
  • It is practically impossible for the die to land on none of the six sides when rolled. (How do I know? Let's say I rolled it 10,000 times. I never observed which side it landed on, but I did observe that it always landed on a side.)

Having no other information, should I assume that the probability of landing on any given side is 1/6? It seems like we should, but I'm not sure how this stuff works.

Edit: Let's assume that I've never seen or heard of dice before, so I don't have cultural information telling me that dice are usually fair.

Version 2

A frog is sitting in front of me on a lily pad at the park. I know from an interpretive sign at the park that this species of frog is capable of croaking. However, having a poor biology education and having never encountered a frog before, I have no information at all about how frequently they croak or what circumstances invoke or prevent croaking. So, for a given period of time, we have two possibilities: at least one croak or no croaks.

Having no other information, should I assume that the probability of the frog croaking at least once during the given time period is 1/2? This seems less intuitive to me than the die example above, but it also seems right.

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This is not a mathematical question, so will probably be flagged as off topic here. It's really a philosophical question about the meaning of "probability".

I think your distinction between the two questions is correct. In the first, you do have prior information: you've seen dice before, and the ones you've seen have been fair (probably - but's that's a hard assertion to justify too). So it's reasonable to assume as a first guess with no further information that this one is fair too.

Response to the edit:

You still have prior information if you've seen cubes before. If the die isn't cubical or you're from some other planet with strange physics that doesn't honor symmetry then you have less prior information and less reason to assume 1/6 for each side.

You confess ignorance of frogs, so there's essentially no prior. The guess of 50% to croak in a minute makes no sense since you could just as logically guess 50% to croak in 10 minutes. I think you need to wait for at least one croak to estimate the croaking frequency.

You might want to read about Bayesian inference. Start here: https://en.wikipedia.org/wiki/Bayesian_inference

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  • $\begingroup$ Let's assume that I have never encountered dice before, either. $\endgroup$ – graumand Mar 22 '16 at 19:02
  • $\begingroup$ You still have prior information if you've seen cubes before. If the die isn't cubical or you're from some other planet with strange physics that doesn't honor symmetry then you have less prior information and less reason to assume 1/6 for each side. Note: discussions like this in comments bother the moderators on this site, so we shouldn't go back and forth often. $\endgroup$ – Ethan Bolker Mar 22 '16 at 19:05

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