I occasionally see puzzles that include language like this:
You go to buy a specific car, whose fair price we’ll call N. You have absolutely no idea what N is
Given any three random integers — X, Y and Z — what are the chances that their product is divisible by 100?
This feels troubling to me, for reasons I talked about here in comments a few years ago but had trouble explaining properly. The thing is that mathematically, there's no way to pick a random integer (or whole number, or real number) such that any integer is as likely as any other one. I'll cite another stackexchange here rather than go into it in detail. You can pick an unbounded random integer, but the way you do it will matter to the answer of the puzzle. For example, you could flip a coin until it gets heads and count how many flips, then flip once more to determine positive or negative. The puzzle creator for the random integer puzzle would hate that idea, though, since he probably wanted us to assume that each integer had a 1 in 5 chance of being divisible by 5, for example, and an integer generated that way has a much less than 1 in 5 chance. A random distribution that gives the intuitive property relevant to that problem would be to pick a power of 100 using the coinflip method, then pick a number less than it by rolling a 100^N-sided die, then flipping a coin for the sign. But that kind of random distribution gives a bad, or non-intuitive, outcome, for the other two problems I linked. For example, if the fair price of a car was generated that way in the first problem, the optimal stopping strategy would take into account that there's a higher probability that the fair price is
So I'm tempted to say that puzzles shouldn't use terms like "a random number" without specifying either a range or a probability distribution, and equivalently shouldn't say you have "no information" about a number. But with puzzles that do that, I do feel like there's almost always consensus about the answer, that somehow we can safely assume the details of the probability distribution to be irrelevant. Is there some way of formalizing this intuition mathematically to "rescue" this sort of puzzle?