Is this a meaningless math question?

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Is this a meaningless math question?

Postby Aelfyre » Tue Feb 15, 2011 8:20 pm UTC

What is the probability that a number chosen randomly from the set of all real numbers is prime?
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Re: Is this a meaningless math question?

Postby theorigamist » Tue Feb 15, 2011 8:33 pm UTC

It's not a meaningless question, and the answer is 0. In fact, the probability that a number chosen at random from the set of all real numbers is RATIONAL is also 0, because the rationals form a set of measure 0.

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Re: Is this a meaningless math question?

Postby Token » Tue Feb 15, 2011 8:37 pm UTC

It *is* a meaningless question - or at least, it's a poorly defined one. What do you mean by "choose randomly"? There are many possibilities, and the intuitive one (choosing uniformly at random) isn't possible.
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Re: Is this a meaningless math question?

Postby gmalivuk » Tue Feb 15, 2011 8:50 pm UTC

Aelfyre wrote:chosen randomly from the set of all real numbers
Chosen randomly how? As Token said, it can't be uniform, which would arguably be the most commonly assumed distribution.

I can define a probability distribution by saying the density function is half the standard normal distribution on all the reals, plus there's a 50% chance of getting 42. In that case, the chance of choosing an integer is 1/2.
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Re: Is this a meaningless math question?

Postby NathanielJ » Tue Feb 15, 2011 8:54 pm UTC

The question needs to be massaged to be made mathematically "proper", but Aelfyre almost surely meant to ask what the probability is of picking a prime if you select a uniformly random natural (not real) number. This has the problem that it's impossible to do so uniformly, but the question can be twerked easily enough:

Pick a number uniformly from the set {1,2,3,...,N} and let P(N) denote the probability that the number you picked was prime. What is [imath]\lim_{N \rightarrow \infty}P(N)[/imath]?

The answer is 0, by the way -- something that is relatively straightforward to show using the prime number theorem.
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Re: Is this a meaningless math question?

Postby Aelfyre » Tue Feb 15, 2011 9:21 pm UTC

NathanielJ wrote:The question needs to be massaged to be made mathematically "proper", but Aelfyre almost surely meant to ask what the probability is of picking a prime if you select a uniformly random natural (not real) number. This has the problem that it's impossible to do so uniformly, but the question can be twerked easily enough:

Pick a number uniformly from the set {1,2,3,...,N} and let P(N) denote the probability that the number you picked was prime. What is [imath]\lim_{N \rightarrow \infty}P(N)[/imath]?

The answer is 0, by the way -- something that is relatively straightforward to show using the prime number theorem.


ok so yes I should have mentioned that obviously logistically it would be impossible to design a machine to do this what with there being a finite amount of energy in the universe and what not.. but what you are saying that if a random natural number is taken from that set then it is impossible for that number to come back as 5? or any other number I assume.. is this another way of saying that there is no way to carry out this experiment in reality? I guess I was thinking a bit more abstractly?

or perhaps I am mistaken in assuming that 0 probability means an occurrence is impossible?

EDIT: ok I think the probability thing is what is breaking everyones brainmeats.. :) would a more clearly defined question be "What percentage of natural numbers are prime?"

the reason I was wondering if it were a meaningless question is because there are infinite natural numbers and also infinite prime numbers so I was wondering if asking that question is similar to asking how to divide by zero.
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Re: Is this a meaningless math question?

Postby SlyReaper » Tue Feb 15, 2011 9:33 pm UTC

Aelfyre wrote:or perhaps I am mistaken in assuming that 0 probability means an occurrence is impossible?

That is a mistake, yes. Choose a random integer, and the probability you'll pick 10697 is 0 because there are infinitely many integers to choose from. But it's not impossible for you to choose 10697.
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Re: Is this a meaningless math question?

Postby Xanthir » Tue Feb 15, 2011 9:55 pm UTC

Aelfyre wrote:EDIT: ok I think the probability thing is what is breaking everyones brainmeats.. :) would a more clearly defined question be "What percentage of natural numbers are prime?"

That's basically the same question. The answer is 0%. ^_^

the reason I was wondering if it were a meaningless question is because there are infinite natural numbers and also infinite prime numbers so I was wondering if asking that question is similar to asking how to divide by zero.

Nope, it's just fine. For example, there are infinite even numbers, too, but the percentage of natural numbers that are even is 50%.
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Re: Is this a meaningless math question?

Postby pizzazz » Tue Feb 15, 2011 10:06 pm UTC

Yeah, there's a slightly strange terminology known as "almost" that essentially means, "it's physically possible, but any x>0 is going to be too large." Like, say you have a line on a plane (such as the real line on the complex plane). If you could randomly select a point on that plan, it would be physically possible to select a real number, but since there are infinite (in fact uncountably so) points in the plane with non-0 imaginary part, any positive real will be too large of a probability. Wikipedia has a better explanation.

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Re: Is this a meaningless math question?

Postby Aelfyre » Tue Feb 15, 2011 10:39 pm UTC

SlyReaper wrote:
Aelfyre wrote:or perhaps I am mistaken in assuming that 0 probability means an occurrence is impossible?

That is a mistake, yes. Choose a random integer, and the probability you'll pick 10697 is 0 because there are infinitely many integers to choose from. But it's not impossible for you to choose 10697.


that is why I was wondering if this was another flavour of dividing by zero since 0 probability not equating to 0 probability is an apparent paradox.

My working definition for probability was a value from 0 to 1. 0 meaning absolute impossibility and 1 meaning absolute certainty.
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Re: Is this a meaningless math question?

Postby Dason » Tue Feb 15, 2011 10:58 pm UTC

Aelfyre wrote:My working definition for probability was a value from 0 to 1. 0 meaning absolute impossibility and 1 meaning absolute certainty.

An impossible event does have probability of 0. But just because an event has a probability of 0 doesn't make it impossible.
A certain event does have a probability of 1. But just because an event has a probability of 1 doesn't mean that it's the only possible event to occur.
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Re: Is this a meaningless math question?

Postby Dopefish » Tue Feb 15, 2011 11:04 pm UTC

Strange things (can) happen when you deal with infinity, zero's cousin without actual number status. Occasionally you get what one might niavely expect, but at least as often you end up with apparent paradoxes.

Assuming time extends infinitely forward, the probability of someone being born at any particular time (or period of time) is zero, but obviously people are born.

Of course, you can flip questions around, and have things with probability 1, that aren't absolutely certain to happen. For example...flip a fair coin an infinite number of times, what is the probability you get at least one heads? I believe the technical term is that it 'almost surely' will happen, but math isn't my subject so there might be some more subtle details surrounding that.

edit: Yup, looks like I used the term right, as it effectively gives my example.

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Re: Is this a meaningless math question?

Postby skullturf » Tue Feb 15, 2011 11:28 pm UTC

For a "typical" continuous random variable, the probability of any one particular value is 0. This may seem strange, but as others have said, "probability 0" does not mean "impossible to occur".

Here's one attempt to make it a little more intuitive.

Consider a very long highway, such as the Trans-Canada Highway, and model it mathematically as a long line segment. Say the random event we're interested in is the location of the next flat tire on the highway. Indicate a particular point on the highway, and ask "What's the probability that the flat tire happens at that point?

Maybe the answer of 0 makes a little more sense if you think of the question as "What's the probability that the next flat tire occurs at this little speck that's less than a hundredth of a millimeter?"

It's also similar, in a way, to questions like "What's the area of a line segment?" or "What's the volume of a line segment or a planar polygon?" The area of a line segment is 0, because it's too small to count, area-wise. In fact, it's the wrong "kind" of thing to count, area-wise. But a line segment still exists. We just assign it an area of 0.

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Re: Is this a meaningless math question?

Postby Xanthir » Tue Feb 15, 2011 11:33 pm UTC

Aelfyre wrote:
SlyReaper wrote:
Aelfyre wrote:or perhaps I am mistaken in assuming that 0 probability means an occurrence is impossible?

That is a mistake, yes. Choose a random integer, and the probability you'll pick 10697 is 0 because there are infinitely many integers to choose from. But it's not impossible for you to choose 10697.


that is why I was wondering if this was another flavour of dividing by zero since 0 probability not equating to 0 probability is an apparent paradox.

My working definition for probability was a value from 0 to 1. 0 meaning absolute impossibility and 1 meaning absolute certainty.

Dason got it. An impossible event has a probability of 0, but having a probability of 0 doesn't make something impossible. A certain event has a probability of 1, but having a probability of 1 doesn't make something certain.

It's easiest to think about it in terms of limits, like NathanielJ talked about earlier in the thread. You can find the probability of randomly selecting a prime among any finite range from 0 to N. As N gets larger, what does this probability tend towards? Identically, as N gets large, what percentage of the numbers in the range are prime? The answer is 0% in both cases - if you try to say it's any positive value, no matter how small, I can name a sufficiently large N that drops the density of primes below the value you gave.

We can look at this with a similar, simpler example. Say I have a set constructed as [1,10,100,1000,10000,...]. There are clearly an infinite number of values in this set. What percentage of positive integers are part of our set? Well, if we look at the range 0-1, the percentage is 1/2. For the range 0-10, it's 2/11. For 0-100 it's 3/101. For 0-1000 it's 4/1001. As you can see, this number will keep getting smaller and smaller, tending towards 0. When you expand the range to the whole of the positive integers, the percentage is thus 0%.
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Re: Is this a meaningless math question?

Postby the tree » Wed Feb 16, 2011 11:52 am UTC

Aelfyre wrote:EDIT: ok I think the probability thing is what is breaking everyones brainmeats.. :) would a more clearly defined question be "What percentage of natural numbers are prime?". The reason I was wondering if it were a meaningless question is because there are infinite natural numbers and also infinite prime numbers so I was wondering if asking that question is similar to asking how to divide by zero.
There are also, say, an infinite number of even numbers but there's no points for guessing that their natural density is 50%, your rephrasing is perfectly meaningful, just not too useful.

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Re: Is this a meaningless math question?

Postby Aelfyre » Wed Feb 16, 2011 1:08 pm UTC

the tree wrote:
Aelfyre wrote:EDIT: ok I think the probability thing is what is breaking everyones brainmeats.. :) would a more clearly defined question be "What percentage of natural numbers are prime?". The reason I was wondering if it were a meaningless question is because there are infinite natural numbers and also infinite prime numbers so I was wondering if asking that question is similar to asking how to divide by zero.
There are also, say, an infinite number of even numbers but there's no points for guessing that their natural density is 50%, your rephrasing is perfectly meaningful, just not too useful.



well yes.. I purposely avoided asking about even numbers since the answer is intuitively obvious ie 50%.. but then your teacher asks you to "show your work" and you end up stabbing her in the face with a Sharpie (very misleading name btw) and then they switch your math class to *another* period of PE...

EDIT: Having thought about this I guess it doesn't fall into the "divide by zero" category... it more appropriately fits in the ".999... = 1" category
An Infinitesimally small but non zero quantity.. you think they'd make a term for that.
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Re: Is this a meaningless math question?

Postby the tree » Wed Feb 16, 2011 1:44 pm UTC

Zero. Definitely zero.

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Re: Is this a meaningless math question?

Postby firechicago » Wed Feb 16, 2011 4:13 pm UTC

Aelfyre wrote:EDIT: Having thought about this I guess it doesn't fall into the "divide by zero" category... it more appropriately fits in the ".999... = 1" category
An Infinitesimally small but non zero quantity.. you think they'd make a term for that.

There have been attempts to construct an arithmetic that includes infinitesimal values. The problem is that you have to handle infinitesimals pretty carefully or they irreparably break a whole bunch of the nice rules that real numbers and their arithmetic obey. Check out the hyperreals if you want to know more.

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Re: Is this a meaningless math question?

Postby supremum » Wed Feb 16, 2011 8:15 pm UTC

SlyReaper wrote:
Aelfyre wrote:or perhaps I am mistaken in assuming that 0 probability means an occurrence is impossible?

That is a mistake, yes. Choose a random integer, and the probability you'll pick 10697 is 0 because there are infinitely many integers to choose from. But it's not impossible for you to choose 10697.

Ummm...no. There is no uniform distribution on the integers because of countable additivity. If you mean pick non-uniformly, there is no reason that probability has to be 0.

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Re: Is this a meaningless math question?

Postby NathanielJ » Wed Feb 16, 2011 10:37 pm UTC

supremum wrote:
SlyReaper wrote:
Aelfyre wrote:or perhaps I am mistaken in assuming that 0 probability means an occurrence is impossible?

That is a mistake, yes. Choose a random integer, and the probability you'll pick 10697 is 0 because there are infinitely many integers to choose from. But it's not impossible for you to choose 10697.

Ummm...no. There is no uniform distribution on the integers because of countable additivity. If you mean pick non-uniformly, there is no reason that probability has to be 0.


Isn't it fair to assume he was talking within the limit of uniform distributions set-up introduced earlier in the thread?
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Re: Is this a meaningless math question?

Postby skullturf » Wed Feb 16, 2011 10:45 pm UTC

We can say that the one-element set {10697} has natural density zero, and we could loosely paraphrase that by saying something like "zero percent of natural numbers are equal to 10697".

But yes, we should be careful about saying "choose a random integer" -- it's impossible to assign a probability to every integer in a way that's both uniform and countably additive.

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Re: Is this a meaningless math question?

Postby Aelfyre » Thu Feb 17, 2011 6:09 pm UTC

firechicago wrote:
Aelfyre wrote:EDIT: Having thought about this I guess it doesn't fall into the "divide by zero" category... it more appropriately fits in the ".999... = 1" category
An Infinitesimally small but non zero quantity.. you think they'd make a term for that.

There have been attempts to construct an arithmetic that includes infinitesimal values. The problem is that you have to handle infinitesimals pretty carefully or they irreparably break a whole bunch of the nice rules that real numbers and their arithmetic obey. Check out the hyperreals if you want to know more.


that is incredibly helpful thank you.. at least there is a term for it.. :)
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Re: Is this a meaningless math question?

Postby charonme » Thu Feb 17, 2011 7:16 pm UTC

The original question is not meaningless in math terms and the answer has repeatedly been pointed out as being zero. In terms of an actual physical world however it is meaningless. There are no "(infinite) sets", "primes", "numbers", "picking primes", "picking elements from (infinite) sets" etc. in the physical world. Some even doubt it is meaningful to speak of numeric probabilities of one-time events within the actual world.

Aelfyre wrote:0 probability means an occurrence is impossible?
This however might be meaningless if you don't define "impossible" in sound math terms. If by "impossible" you mean that there is a logical contradiction somewhere in the idea, then no, it doesn't mean it's impossible. If by "impossible" you mean that the probability is equal to zero (my slightly preferred definition), then obviously yes, probability=0 means it is impossible.

But if you're asking about the relationship between a mathematical probability being zero and the possibility of the actual event occurring in the real world, then this is meaningless.

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Re: Is this a meaningless math question?

Postby Charlie! » Thu Feb 17, 2011 7:24 pm UTC

charonme wrote:Some even doubt it is meaningful to speak of numeric probabilities of one-time events within the actual world.

Well, those people are silly.

But yea, it highlights an incompleteness in how "possible" is defined.
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Re: Is this a meaningless math question?

Postby gmalivuk » Fri Feb 18, 2011 6:24 pm UTC

charonme wrote:The original question is not meaningless in math terms and the answer has repeatedly been pointed out as being zero.
No, because as worded the original question doesn't give us enough information to pick an answer.
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