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.)
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.
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.