Theoretical probability question

[Signal 11]

The number of ways to choose 5 items out of a pool of 10

At least 5 though, not exactly 5.

 

[xenon]

The number of ways to choose 5 items out of a pool of 10 is given by the formula 10!/5!x5! = 252

So the chance of getting it right at random is 1/252

So it's not a great lock. A bit better than a normal 3 digit combination, but not by much.

Thanks. I will take another look on my computer later, I don’t actually know what the ! symbol denotes.

 

[xenon]

At least 5 though, not exactly 5.

For the purpose of the question I have assumed must have five.

 

[Signal 11]

so 1 + 10 + 10x9 + 10x9x8 + 10x9x8x7 + 10x9x8x7x6 = 36100?

forgot to divide...
10!/10! + 10!/9!x1! + 10!/8!x2! + 10!/7!x3! + 10!/6!x4! + 10!/5!x5! = 638

 

[Signal 11]

Thanks. I will take another look on my computer later, I don’t actually know what the ! symbol denotes.

Factorial - Wikipedia

 

[kabbes]

... And just to prove my point, here are all 252 possible combinations:

 

[TruXta]

Slow day at work, kabbes boy?

 

[Wilf]

"How can walking into a door not result in bumping into a door? That's what walking into a door means."

Except when the door is in the opened state.

Now imagine instead of doors you've got walls. And instead of open/closed you've got "are my atoms lined up in a very specifc way that they won't interact with the wall, allowing me to pass through it?"

Of course the chances of the door being open are many, many times greater than the chance of your atoms being lined up in the correct way to pass through the wall. But they're roughly the same thing

if I walk into a wall the result will be me bumping into a wall.

Aren't walk into a wall and bump into a wall synonyms, at least in this context? They both, in themselves, indicate contact with the wall, otherwise it would be walk through a wall. As set out in the OP, they are the same event, there is no causation, no probability.

 

[kabbes]

Aren't walk into a wall and bump into a wall synonyms, at least in this context? They both, in themselves, indicate contact with the wall, otherwise it would be walk through a wall. As set out in the OP, they are the same event, there is no causation, no probability.

What do you understand by the word "contact"?

 

[Wilf]

What do you understand by the word "contact"?

Look, I'm playing the country bumpkin role on this thread, don't take me out of my comfort zone.

 

[Fez909]

Aren't walk into a wall and bump into a wall synonyms, at least in this context? They both, in themselves, indicate contact with the wall, otherwise it would be walk through a wall. As set out in the OP, they are the same event, there is no causation, no probability.

Is walk into a house and bump into a house the same thing?

 

[Wilf]

Is walk into a house and bump into a house the same thing?

No, thus my wording ('in this context'). Something like Roddy Doyle's 'the woman who walked into doors'. 'Walked into', with reference to walls, denotes physical contact with a hard surface, not some sort of transition/passage through it.

 

[kabbes]

No, thus my wording ('in this context'). Something like Roddy Doyle's 'the woman who walked into doors'. 'Walked into', with reference to walls, denotes physical contact with a hard surface, not some sort of transition/passage through it.

That works when we're talking about the colloquial level at which we experience everyday life. It's trickier when we start to get into the difficulties of quantum events. You have to be quite precise about concepts such as "physical contact" and what that actually means.

 

[kabbes]

Slow day at work, kabbes boy?

Pshaw. I can generate 252 combinations with one hand whilst I crank a calculator handle with the other.

 

[Wilf]

That works when we're talking about the colloquial level at which we experience everyday life. It's trickier when we start to get into the difficulties of quantum events. You have to be quite precise about concepts such as "physical contact" and what that actually means.

17000 posts and I'm not changing now.

Well, yes, I know. I just like blustering through some of these problems. For example the logic problem of 2 identical twins, one always lies, one always speaks the truth... what you question would you ask to determine which brother you are speaking to? Answer: ask him if he's an Axminster carpet.

 

[NoXion]

17000 posts and I'm not changing now.

Well, yes, I know. I just like blustering through some of these problems. For example the logic problem of 2 identical twins, one always lies, one always speaks the truth... what you question would you ask to determine which brother you are speaking to? Answer: ask him if he's an Axminster carpet.

That's just an intellectual puzzle though, whereas the original post treads on the territory of physics somewhat. One can't bluster one's way through physics, you might as well be trying to control the weather by cursing.

 

[Jonti]

mrs quoad — when you ask if probability is “theoretical”, are you asking if probability theory itself is like a scientific theory (ie hypothesis that appears to be correct based on empirical evidence) as opposed to mathematical theory (ie tautologous statements derived logically from axioms)?

Because if so, no it is not like a scientific theory. It is a construct deriving from something called measure theory, which is a well-definited and rigorous branch of maths. It gives you the tools to define the measurement of sets in such a way that you can apply the same kind of approach to likelihood than you do to other topologies. From that Wiki article:

• Every probability space gives rise to a measure which takes the value 1 on the whole space (and therefore takes all its values in the unit interval [0, 1]). Such a measure is called a probability measure. See probability axioms.

If, on the other hand, you’re asking if the application of probability in the real world is theoretically valid, the answer is that it is as valid as any other application of a theoretically pure construct. You really need to differentiate risk (the known unknowns) from uncertainty (the unknown unknowns), or process error (shit happens) from parameter error (but maybe not in the way envisaged) from model error (and what was THAT shit about?). You also need to accept model limitations (ie we know it isn’t perfect but use it anyway because meh, close enough).

These issues, though, can also be tackled with the tools of probability. Parameter uncertainty is dealt with via Bayesian approaches. Model uncertainty has its own mathematical branch. The implications are understood within a rigorous theoretical framework, not just accepted through empirical evidence.

Just wanted to point out that although tautologous statements derived logically from axioms does describe a lot of maths, it doesn't describe it all by any means. Recall that Godel showed that doesn't apply to arithmetic; he showed that arithmetic is not axiomatisable. I wonder if that is connected to how fundamental arithmetic is to how things work in the real world.

Anyway, that amazing fact convinces mathematician Roger Penrose that a computer (which is limited to tautologies) is nothing like a human mind, and that the quest for strong AI is misconceived.

 

[kabbes]

I thought this was the axiomatic basis for arithmetic?

Peano axioms - Wikipedia

It’s been 20 years admittedly, but I thought Godel showed you couldn’t get beyond these axioms rather than that there were no axioms. Incompatibility of completeness and consistency, in other words.

Not my strong point though, I have to say.

 

[littlebabyjesus]

Just wanted to point out that although tautologous statements derived logically from axioms does describe a lot of maths, it doesn't describe it all by any means. Recall that Godel showed that doesn't apply to arithmetic; he showed that arithmetic is not axiomatisable. I wonder if that is connected to how fundamental arithmetic is to how things work in the real world.

Anyway, that amazing fact convinces mathematician Roger Penrose that a computer (which is limited to tautologies) is nothing like a human mind, and that the quest for strong AI is misconceived.

Sort of. You're referring to The Emperor's New Mind, no? Penrose uses Goodstein's theorem as his illustration of the point, and irrc Goodstein's theorem can be taken to be a Godel statement for arithmetic.

But we need to be careful about what Godel showed. He showed that any axiomatic system must contain a statement that you know to be true but that cannot be proved by the system alone. Penrose's preoccupation is with this 'knowing' and what it might mean. He still hasn't got to the bottom of it!

 

[Jonti]

I thought this was the axiomatic basis for arithmetic?

Peano axioms - Wikipedia

It’s been 20 years admittedly, but I thought Godel showed you couldn’t get beyond these axioms rather than that there were no axioms. Incompatibility of completeness and consistency, in other words.

Not my strong point though, I have to say.

It's not that there are no axioms, it's more you need an infinite number of them. So to speak, you have to add a new axiom every time you come to a statement that you know to be true but that cannot be proved from the existing axioms alone. There's no end to this process.

 

[Jonti]

Sort of. You're referring to The Emperor's New Mind, no? Penrose uses Goodstein's theorem as his illustration of the point, and irrc Goodstein's theorem can be taken to be a Godel statement for arithmetic.

But we need to be careful about what Godel showed. He showed that any axiomatic system must contain a statement that you know to be true but that cannot be proved by the system alone. Penrose's preoccupation is with this 'knowing' and what it might mean. He still hasn't got to the bottom of it!

Steady on, not all formal systems are incomplete (my emphasis added)

In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems; otherwise the system is said to be incomplete. The term "complete" is also used without qualification, with differing meanings depending on the context, mostly referring to the property of semantical validity. Intuitively, a system is called complete in this particular sense, if it can derive every formula that is true. Kurt Gödel, Leon Henkin, and Emil Leon Post all published proofs of completeness.

 

[littlebabyjesus]

Steady on, not all formal systems are incomplete (my emphasis added)

They're all either incomplete or contain a statement that is true but cannot be proved using just that system. A few years ago I could have given you an outline of the mathematical proof of it, but I'd have to look it up now. But it's essentially a statement of the same thing as the stopping problem of the Turing machine. There will be a statement in there that the system cannot decide upon, and it's normally of a self-referential character. Put in terms of a Turing machine, it will just keep going for ever without ever deciding whether or not the statement is true. But very often it is true. We can see that it is true. We can know that it is true. Which is Penrose's point.

Not sure what your quote is referring to out of context.


There are people, such as Max Tegmark, who put forward the idea that the Universe is not just described by maths. It is maths. I've been resistant to the idea, but I'm coming round to it. If it is the case, then the Godel statement of such a system might very well be something like 'the Universe is mathematics'.

 

[Jonti]

The quote's from the wikipedia entry for completeness. I'm very rusty on all this stuff, but isn't "containing a statement that is true but cannot be proved using just that system" the definition of incompleteness? being as the set of axioms is never complete.

I do think Penrose has a point. Daniel Dennett would disagree and snort about sky hooks.

 

[littlebabyjesus]

The quote's from the wikipedia entry for completeness. I'm very rusty on all this stuff, but isn't "containing a statement that is true but cannot be proved using just that system" the definition of incompleteness? being as the set of axioms is never complete.

I do think Penrose has a point. Daniel Dennett would disagree and snort about sky hooks.

I like Dennett. I liked his book about evolution. His book Consciousness Explained doesn't explain consciousness. I also think Penrose has a point. I still don't quite get Dennett's denial of the validity of the concept of qualia.

 

[Jonti]

Well, one can, without formal contradiction, deny subjectivity. So he does, to prove how clever he is

 

[Jonti]

...
There are people, such as Max Tegmark, who put forward the idea that the Universe is not just described by maths. It is maths. I've been resistant to the idea, but I'm coming round to it. If it is the case, then the Godel statement of such a system might very well be something like 'the Universe is mathematics'.

Maths is everything, eh? That goes beyond Galileo's “Mathematics is the language in which God has written the universe”. But how is "yellow" in any way a mathematical entity?

 

[kabbes]

It's not that there are no axioms, it's more you need an infinite number of them. So to speak, you have to add a new axiom every time you come to a statement that you know to be true but that cannot be proved from the existing axioms alone. There's no end to this process.

Surely that still means, though, it is a system of axioms and tautologous statements derived from those axioms. I don’t understand what your objection was to my characterisation of maths as being a series of tautologous statements derived logically from axioms. If you’re not saying arithmetic is axiom-free, are you saying it has statements that are neither axiomatic nor derived logically from those axioms?

 

[Jonti]

Yes, that's what I understand. But I've just ordered Nagel and Newman's excellent Gödel's Proof to brush up on things. I last read it quite some time ago.

 

[mauvais]

Maths is everything, eh? That goes beyond Galileo's “Mathematics is the language in which God has written the universe”. But how is "yellow" in any way a mathematical entity?

It's a wavelength, for a start. Probably not the best example.

 

[Jonti]

I was thinking of the sensation of seeing the colour yellow. Wavelength as such has no colour.