Every whole number can be mapped onto a number divisible by 10.You've lost me!
1 maps to 10, 2 to 20 etc
Theoretical probability question
- Thread starter Corax
- Start date
1 maps to 10, 2 to 20 etc
Corax
Luke 5:16
Okay.
Erm. Sorry Corax about the slight hijack but thought about this earlier and doesn't seem worth a new thread...
Combination locks. The type you get on padlocks, with 10 switches. So, 10 to the power 2, gives us 1024 possibilities. 2^10.
But how do I work out how many combinations there are if, the code has to have at least n digits. That is, use at least n switches? Say, 5 for example?
No, I've not forgotten the combination but thought, what if I did...
Combination locks. The type you get on padlocks, with 10 switches. So, 10 to the power 2, gives us 1024 possibilities. 2^10.
But how do I work out how many combinations there are if, the code has to have at least n digits. That is, use at least n switches? Say, 5 for example?
No, I've not forgotten the combination but thought, what if I did...
Cool. So if there are no spare whole numbers unaccounted for, there must be at least as many numbers divisible by 10 as there are whole numbers.
Corax
Luke 5:16
For numbers 1...10, there's only one number neatly divisible by ten, and 10 whole numbers. So the infinite set of whole numbers is bigger than the infinite set of multiples of 10, yes?
mauvais
on reddit or something
Unless I'm misunderstanding your scenario, it's just 10^n, isn't it. So this:
is 10^4 = 10000 which is self-evident anyway.
No!
Corax
Luke 5:16
I'm being thick, aren't I?
littlebabyjesus
one of Maxwell's demons
How big is infinity?
I think you're making the mistake of thinking of infinity as a number. It isn't. It's really an incompletely defined thing. What kabbes is talking about is called a difference between countability and uncountability. Cantor's diagonal argument, which I linked to earlier in the thread, is pretty cool. I don't think it's that hard to see how it works.
littlebabyjesus
one of Maxwell's demons
I'm not kabbes, but I can answer that. No, it isn't. But probability is an expression of incomplete knowledge. It is a quantification of incomplete knowledge.
Which isn’t theoretical?
littlebabyjesus
one of Maxwell's demons
What do you mean by 'theoretical'? No it's not. Take quantum mechanics. Schrodinger's equation gives you the complex number probability amplitude of a particular quantum state. That's extremely precise: repeat the experiment many times or make lots of measurements and the distribution of results will match the calculation really really really precisely. That's not just theoretical - it's an accurate description of the state (as far as the knowledge of that state goes).
Unless I'm misunderstanding your scenario, it's just 10^n, isn't it. So this:
is 10^4 = 10000 which is self-evident anyway.
Ah right, so, my padlock requiring a 5 digit combination, has 10000 possible combinations. (4^10 not 5^10 because excluding 0?)
I am a bit tired TBH.
That video is completely correct and you are describing the exact opposite of what that video says.
Sorry!
And that isn’t theoretical?
I mean.
It sounds as if the observations match the theory really well.
That’s still theoretical, though. Isn’t it?
A better theory could come along.
Corax
Luke 5:16
But you can count the whole numbers between and 10.
And you can count the number of x.x numbers between 1 and 10.
And the second set is bigger than the first set.
So no matter how far you extend it, even to infinity, the infinite set of x.x numbers will be bigger than the infinite set of whole numbers.
No?
littlebabyjesus
one of Maxwell's demons
It's right. That's cantor's diagonal argument.
Corax
Luke 5:16
Okay, thanks. That's what I was trying to describe, anyway!
Oh my God, 11pm on a school night is NOT the time to begin that discussion!
What we call “probability” in reality describes at least two distinct categories of thing. Purely mathematical constructs, which are theoretically pure, and statements about the extent of our knowledge, which are not really probabilistic events at all, frankly.
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littlebabyjesus
one of Maxwell's demons
You have a point. Probability only makes sense as a prediction about something that hasn't happened yet.
Tbf, I think what this boils down to is “theory is supported by observations therefore it isn’t theory.”
Which.
Yeah.
littlebabyjesus
one of Maxwell's demons
Corax
Luke 5:16
That would be an ecumenical matter.
