Talk:Classical Probability is Probability Measure

It occurs to me that the proof of the third axiom depends on the fact that the events are disjoint. So I browsed back to Classical Probability Model and couldn't find anywhere this was stated. I did find it in the statement of the definition of Probability Measure, though, but this time (as a naive reader) I've lost the track of where the assumption of disjointness comes.

In a proof like this where we are referring to a set of axioms, and in the proof we refer to another proof, all you've got is "it fulfis (x) axiom because of (y) result." I'm becoming unhappy with this approach (I've used it a lot in the past - in fact I think I still do) and I wonder whether it needs something more:

"Third axiom states that: blah blah. X result states that: yada yada. This-object fulfils the conditions for X result to hold, because yak blether. Therefore X result can be applied to this-object and so the third axiom is seen to hold."

I started doing that for some of the fiddly topological properties where a property holds merely by the interpretation of the definition of the property as applied to the object as defined. The proof in these cases is a one-liner, but the hard work is i explaining what the definition means in the context of the given page.

What does anyone else think? --prime mover 03:28, 11 December 2011 (CST)

I'll think about this. Do you know the difference between mutually exclusive events and pairwise disjoint...um, things? (I don't.) It would help me address the issue if I was more clear on the difference, I think. --GFauxPas 07:10, 11 December 2011 (CST)

That's the whole point. The Classical Probability Model says nothing about the mutual exclusivity of the events. Whether this is something which is to be deduced or not I can't tell, as I don't have access to your source works (and have absolutely no desire to immerse myself in probability theory at the moment to do research). Whether the mutually exclusiveness is to be deduced as a consequence of something else or not I don't know. But at the moment there appears to be no such causal link drawn.

"um, things" you mean "sets". An "event" is an element of an "event space", which is a strategically defined subset of the powerset of the sample space, and so is an element of the power set of the sample space. Mutually exclusive events are disjoint sets in the event space. --prime mover 09:30, 11 December 2011 (CST)

Can we use something like the Pigeonhole Principle to prove that the amount number of events is exactly the same as the cardinality? Something like, the amount number of events can't be more than $\#$ because <reason> Similarly, the amount number can't be less because <reason>. --GFauxPas 07:52, 11 December 2011 (CST)

I don't know. Can you? (BTW don't use "amount", use "number". "Amount" is used for uncountable (in the natural language sense) quantities, like "beer" and "cheese", "number" is used for things you can count, like "apples" or "pints of beer" or "cheese sandwiches".) --prime mover 09:30, 11 December 2011 (CST)

I apologize for my grammar, thank you for correcting me. --GFauxPas 14:45, 11 December 2011 (CST)

It only matters on the main pages. --prime mover 16:14, 11 December 2011 (CST)

BTW I'm absolutely in no way saying that you are responsible for working on this proof, I will continue thinking about it, but in terms of "access to my source works" the link to the proof is at the bottom of the page, pages 62 and 25 in the pdf. But you might be disappointed, and I'm sure you are quite busy as it is. --GFauxPas 17:01, 11 December 2011 (CST)

D'oh! So it is. I'll take a look at it properly in due course. --prime mover 17:08, 11 December 2011 (CST)