[EDIT] I think I understood joefromchicago's way of putting it, but there is more to it if you read below. I think I erred for abstracting even the actual numbered balls, like thinking of 49x48x... as distinct but variable at the same time picks... I think I probably invented a mental bias of my own
... or because it's friday night
but that's how I build up my knowledge base.[/EDIT]
Yes, I understood that all along. What I don't understand is that by analogy, applying a proportion to a number is applying a fraction that represents that proportion, like 1/4 is clear I take 25% out of a number.
I acknowledge that order doesn' t matter, because I observed that, that I need to divide by the number of permutations.
I do not understand the inner working requirement however, or reasoning abstracting the fact I know the formula and observed the property that order doesn't matter.
It's as knowing what gravity does, when we don't know how it works...
It's because that lack of understanding I got no answers to this yet:
The understanding I'm talking of is the difference between formal and self education.
The key seems to stand in the fact that dividing n! by the number of combinations of numbers taken from it, practically unravels the basic partition w/o permutations from the n! as if "substracting" from the total possibilities, just in the way dividing by (n-r)! is "substracting" the repetitions out of all possibilities. So in other words I'm removing properties from all posibilities by dividing.
Like I said, the flaw in the wiki "lottery mathematics" is the way in which it is explained. They came up with permutations at first, or arangements how we call them. In my country they taught us that permutations are n!, n!/(n-r)! are arangements and n!/r!(n-r)! are combinations. Then they "substracted" the fact order is not important, because the number they came up with is describing just one arrangement, true. It would've been just as easy for me to assess that because a ticket has no order meaning and no repetition, I should apply the combinations formula directly.
However, in order to solve a complex problem, one need to be intimate with the meaning of the objects and operations meaning. For instance I know that in http://able2know.org/topic/175253-1
I need to apply (n+r-1)!/r!(n-1) to solve the probability of 4 out of 6, but not only I do not understand why I divide one thing by another, that won't help solve the problem because it's tied to the probability of guessing those 4 with another 5.
Then there's the another problem: take two dice, one is the lottery, one is the player. The chance the lottery and the player pick the same number is 1/6 * 1/6 = 1/36 not 1/6. If you don't like the player and lottery example, try two random generators, it's the same thing. So the probability of winning the lottery is not 1:14 million. That's just the probability of a particular outcome of the lottery alone it would seem. Dr. Math says it's not though no mathematical demonstration is given. Statistically I haven't tried for 6/49 probability, though all the other distributions seem to model the statistical reality pretty close, I would say the probability for winning 6/49 is lower than 14 million, speaking from experience. The math just doesn't make sense.