# Definition of [probability]

The short definition of is:

A mathematical puzzle that involves randomness.

Is that really what probability is, though? Surely not everything that's random is probability. For example, Can prove that any random numbers between 1-100 can be divided into two equal partially sum groups? [sic]. Though here randomness is being confused with arbitrariness, I think that the definition could do with a bit of clarifying anyway.

But I'm not an expert on probability, as evidenced by my false answer to the recent Elchanan Mossel’s dice puzzle.

Does the community think that this is a fine definition of , and if not, what should we change it to?

For reference, the long definition is:

A probability puzzle is a puzzle, usually mathematical, that involves uncertain outcomes.

Probability puzzles often involve , , or , although other mechanisms may act as sources of uncertain outcomes.

For example:

Unfair coins at South Park Elementary

Another example of a probability puzzle is the Monty Hall problem: the game show problem with three doors, one car, and two goats.

I guess there are three questions here.

1. Is every question involving randomness a probability question?

Obviously not (e.g., imagine a story involving people gambling with dice, where the randomness is just there as scenery), but I think any question about randomness is a probability question, at least if it's quantitative about it. "Mathematical puzzle that involves randomness" isn't too far off, it seems to me.

The "equal partial sums" question linked by boboquack is indeed not a probability question, but boboquack has correctly diagnosed the real problem there: the title uses the word "random" but doesn't really mean "random". I think this question is largely irrelevant to the point at issue.

1. Does every probability question involve randomness?

Maybe not, for two reasons. Firstly, sometimes you use probabilistic methods to solve a not-obviously-probabilistic question (e.g., you want to prove that a Thing with a certain Property exists, and you do it by saying "Consider a randomly chosen Thing; then the probability that it has the Property is at least such-and-such, which is bigger than 0; therefore at least one Thing exists with that Property"). But these probably shouldn't be considered probability questions in our sense.

Secondly, the techniques of probability theory are applicable whenever we have uncertainty, whether or not there is actual randomness. So if I want to decide whether my wife is having an affair, or whether some religion's claims are true, I might do it by making an initial assignment of probabilities and then updating according to the available evidence using Bayes' theorem. But questions of this sort where the uncertainty doesn't come from quantifiable sources of randomness are likely not to have definite answers and therefore not to be suitable for Puzzling.

1. Do the observations above mean there's a problem with how the probability tag is described?

I don't think they do. The objections above are all quibbles, in this context. Unless we have actual examples of questions that are being misclassified and would be classified better if we had a different definition for the tag, I think it's fine.

• In view of your first question here, I've changed the tag wiki excerpt to "... whose essential nature involves randomness". Does that work better, or is it unnecessarily clunky? – Rand al'Thor Sep 14 '17 at 10:40
• Seems OK. (In view of my second question, it's not clear to me that it's an actual improvement, but I think it's harmless. Our tag descriptions are not meant to be academic treatises...) – Gareth McCaughan Mod Sep 14 '17 at 12:04
• And in view of the answer to your third question, who cares anyway :-) – Rand al'Thor Sep 14 '17 at 13:36