Temperature probablity

[austinpoker]

In a statistics class I am taking we discussed the probability of the temperature in a room being 72 degrees. Our teacher said that the probability of it being 72 degrees in the room was 0. This was because the temperature could always be measured more precisely to infinity (78.0000…1). This seems impossible because if you assume that the temperature drops from 73 degrees to 71 degrees then while its dropping it must pass through exactly 72 degrees not (78.00000…1). So since there is a number that is exactly 72 degrees then there must at least some chance (0.0000…1) that the temperature is exactly 72 degrees in the room. After much heated discussion she told us that the probability of it being 72 degrees was 0, not a very small number close to 0. Is this correct or not?

[tyrus72]

It is correct. Probability of being 72.000... is zero. Why? How many other possible temperatures are there? Infinity. What is 1/infinity? Zero.

[Matt R.]

I think your teacher's on crack. Did she provide any type of proof? What was her response to your reasoning (the temp. must pass through exactly 72 as it goes from 73 to 71)? By her logic, the temperature never reaches any exact value.

[tyrus72]

[ QUOTE ]
I think your teacher's on crack. Did she provide any type of proof? What was her response to your reasoning (the temp. must pass through exactly 72 as it goes from 73 to 71)? By her logic, the temperature never reaches any exact value.

[/ QUOTE ]

Wrong. By her logic, the probability of the temp being at any exactly specified temp (to infinite sig figs) is zero. This is absolutely correct. The key is the infinite significant digits.

If you need to rationalize this: Say I have an equal probability of picking any integer at random. What is the probability that I pick 134,006? Answer: 0. This number has a probability of 1/infinity. I can always show the probability is less than any number "close to 0" that you want to specify -- thus it equals zero.

[Matt R.]

[ QUOTE ]
It is correct. Probability of being 72.000... is zero. Why? How many other possible temperatures are there? Infinity. What is 1/infinity? Zero.

[/ QUOTE ]

So for there to be a finite probability, you have to give a range rather than a point?

Edit -- durr. I mean non-zero probability.

This makes sense to me (not saying I know it's right though, lol)

[tyrus72]

[ QUOTE ]
[ QUOTE ]
It is correct. Probability of being 72.000... is zero. Why? How many other possible temperatures are there? Infinity. What is 1/infinity? Zero.

[/ QUOTE ]

So for there to be a finite probability, you have to give a range rather than a point?



[/ QUOTE ]

Correct.

[Matt R.]

[ QUOTE ]
Wrong. By her logic, the probability of the temp being at any exactly specified temp (to infinite sig figs) is zero.

[/ QUOTE ]

Isn't that what I said?

Explain this one though. As to what the OP said, to go from 73 degrees to 71 degrees you much exactly go through 72. Therefore at some point the tempurature is exactly 72 degrees. If it is possible for the temperatue to be exactly 72 degrees, than the P must be >0.

Is there something weird going on with the language "go through 72" that I'm missing?

I see your point in the other post, but I'd like a counter argument for the OP's point that I reiterated here. I think the P=0 is probably right, but it's more of a consequence of an axiom from probability theory that's difficult to convey linguistically. That probably made no sense. Basically I mean P=0 in this case does not mean impossible, but it kind of still does, lol (Edit -- okay, I kind of mean impossible in the sense "to measure temp with absolute precision". But it's still possible for it to exist.... still clear as mud I think). Chalk this up to me not being able to describe a mathematical idea using words.

(anyway, I'd like to hear a counter argument -- just ignore my nonsense at the end)

[tyrus72]

[ QUOTE ]
[ QUOTE ]
Wrong. By her logic, the probability of the temp being at any exactly specified temp (to infinite sig figs) is zero.

[/ QUOTE ]

Isn't that what I said?

[/ QUOTE ]

No, it isn't close to what you said.

[tyrus72]

[ QUOTE ]
As to what the OP said, to go from 73 degrees to 71 degrees you much exactly go through 72. Therefore at some point the tempurature is exactly 72 degrees. If it is possible for the temperatue to be exactly 72 degrees, than the P must be >0....

I see your point in the other post, but I'd like a counter argument for the OP's point that I reiterated here. I think the P=0 is probably right, but it's more of a consequence of an axiom from probability theory that's difficult to convey linguistically. .

[/ QUOTE ]

I already conveyed it "linguistically" and provided a counterargument. And how can something be "probably right" as a "consequence of an axiom" but also wrong because its hard to explain?

If you don't think it is zero, then name a number close to zero that you think it is and I will show you it is less than that number. We can play that game a few times until you eventually concede that 1/infinity is indeed equal to 0 and not just close to 0.

[bobman0330]

Isn't thermal energy quantized?

[Matt R.]

[ QUOTE ]
By her logic, the temperature never reaches any exact value.

[/ QUOTE ]

[ QUOTE ]
Wrong. By her logic, the probability of the temp being at any exactly specified temp (to infinite sig figs) is zero

[/ QUOTE ]

No need to be an ass. At least point out the difference if it's "not even close". To someone not trained in probability theory, these points look exactly the same.

If the probability of the temperature being at an exact point is zero (to an infinite sig figs.), the temperature never actually IS at that point. As an analogy: the probability of rolling a 7 on a six sided die is zero, therefore the die never reaches a value of 7. Elementary? Sure. But this is precisely what it looks like to someone who doesn't know probability theory in depth.

In short, don't be so condescending. If you want to reply at least explain why I'm "not even close".

[tyrus72]

[ QUOTE ]
[ QUOTE ]
By her logic, the temperature never reaches any exact value.

[/ QUOTE ]

[ QUOTE ]
Wrong. By her logic, the probability of the temp being at any exactly specified temp (to infinite sig figs) is zero

[/ QUOTE ]

No need to be an ass. At least point out the difference if it's "not even close". To someone not trained in probability theory, these points look exactly the same.

If the probability of the temperature being at an exact point is zero (to an infinite sig figs.), the temperature never actually IS at that point. As an analogy: the probability of rolling a 7 on a six sided die is zero, therefore the die never reaches a value of 7. Elementary? Sure. But this is precisely what it looks like to someone who doesn't know probability theory in depth.

In short, don't be so condescending. If you want to reply at least explain why I'm "not even close".

[/ QUOTE ]

What is the probability that I pick 7 if I have say an equal probability of picking any integer? Answer: 1/infinity = 0. So does this mean that I can never really pick a number? No.

I don't see how the excerpt above shows me as an ass, at least not as much of an ass as I can be. You asked a question. I answered it matter of factly. Obviously, the temperature always IS at some particular value. That does not mean that the probability of being at that value given an infinite number of possibilities cannot still be zero.

Frankly, I could have said something like "you are the idiot on crack" or something to that effect, which is the tone you used to describe the teacher in question.

[Matt R.]

OK, let's play your game.

If P=0 it is impossible for an event to occur.

I walk 10 yards. It is impossible for me to be precisely 5 yards from my start point to an infinite number of sig figs. Thus the P that I am exactly 5 yards from my start point is P = 0. If I never can make it 5 yards from my start point, I certainly can't make it 10. (Now before you get all high-and-mighty again, I'm sure you mathematically prove this is wrong. I'm just showing why it's non-intuitive.)

You never provided a counter argument. You simply said the probability that it is 72 degrees is zero because there's an infinite number of other possibilities and you can't measure temp to an infinite number of sig figs. If it cannot be 72 degrees, but it can go from 73 to 71, it has to "skip over" 72. This is what's hard to understand about it if you think about it "linguistically", as I said in my other post. Of course it doesn't really skip over it.

Note that I never said 1/infinity is not zero. I'm also not saying you are wrong. I'm saying this is very non-intuitive. And it is very difficult to explain it in words. Hence the original post. [censored].

[madnak]

[ QUOTE ]
If P=0 it is impossible for an event to occur.

[/ QUOTE ]

This can't be true, based on what you're saying.

[madnak]

Excuse my algebraic math, but some things just don't seem right here. i is infinity

1/i = 0

1 = 0i

Huh?

[Matt R.]

lol. I said "I think your teacher is on crack". This is clearly a joke -- I'm poking as much fun at myself because I don't understand it.

Your condescending tone matches up well with your lack of a sense of humor.

You ripped apart my statement that "P=0 does not really mean 'impossible'". Which is why I was saying it is difficult to explain these ideas linguistically. (Note that I qualified I knew the idea wasn't coming across well, yet you still had to point out that I was wrong).

From your example:

[ QUOTE ]
What is the probability that I pick 7 if I have say an equal probability of picking any integer? Answer: 1/infinity = 0. So does this mean that I can never really pick a number? No.

[/ QUOTE ]

It is also clearly possible that you pick 7, even if P = 0. Thus, the idea of P=0 equaling "impossible" is hard to express using words. That was pretty much my point.

And don't worry -- I believe that you're right. There's no need to throw a hissy fit and offer to "play my game to prove me wrong". I simply wanted a little further explanation. Sorry to encroach upon your mathematical throne.

[Matt R.]

[ QUOTE ]
[ QUOTE ]
If P=0 it is impossible for an event to occur.

[/ QUOTE ]

This can't be true, based on what you're saying.

[/ QUOTE ]

madnak,
I know, I was kind of combining the other guy's assumptions and mine to prove a point. Apparently P=0 does not really mean "impossible" the way I typically think of it. I was creating another example to show why it was hard to fathom the temperature question.

I fully acknowledge that I'm probably wrong. I was just trying to get a little bit more of an explanation.

[Matt R.]

OK, I reread my original reply and maybe it sounded like I was implying the teacher was an idiot. But I wasn't. So I apologize if it offended you. I also apologize if you weren't intentially being condescending in your replies, but it really comes across that way when you say things like "wrong, you're not even close" or whatever.

PS -- I also didn't literally mean I think she is on crack, cocaine, heroine or any other type of drug. Just to clarify.

[MathEconomist]

You can't manipulate infinity in this manner.

As for the OP, my formal background on this part of probability theory is kind of shallow, but I think this has something to do with sets of measure zero. Events that happen with probability zero may be 'really impossible' (like rolling 7 on a 6 sided die or getting the number 2 on a random draw from the interval [0,1]) or simply have probability zero because the event in question is actually a set of measure zero and we have a continuous distribution.

So the probability of getting any specific temperature (or any arbitrarily large but finite collection of temperatures) from a random draw is zero. But obviously from any random draw you get SOME realization of the random variable. (Of course whether temperature is actually a continuous random variable is a physical question that I don't know the ansewr to.)

[tyrus72]

[ QUOTE ]
You can't manipulate infinity in this manner.

As for the OP, my formal background on this part of probability theory is kind of shallow, but I think this has something to do with sets of measure zero. Events that happen with probability zero may be 'really impossible' (like rolling 7 on a 6 sided die or getting the number 2 on a random draw from the interval [0,1]) or simply have probability zero because the event in question is actually a set of measure zero and we have a continuous distribution.

So the probability of getting any specific temperature (or any arbitrarily large but finite collection of temperatures) from a random draw is zero. But obviously from any random draw you get SOME realization of the random variable. (Of course whether temperature is actually a continuous random variable is a physical question that I don't know the ansewr to.)

[/ QUOTE ]

Exactly. But I thought if the "simple" explanation went over his head, this one would as well.

[tyrus72]

[ QUOTE ]
lol. I said "I think your teacher is on crack". This is clearly a joke -- I'm poking as much fun at myself because I don't understand it.

Your condescending tone matches up well with your lack of a sense of humor.

You ripped apart my statement that "P=0 does not really mean 'impossible'". Which is why I was saying it is difficult to explain these ideas linguistically. (Note that I qualified I knew the idea wasn't coming across well, yet you still had to point out that I was wrong).

From your example:

[ QUOTE ]
What is the probability that I pick 7 if I have say an equal probability of picking any integer? Answer: 1/infinity = 0. So does this mean that I can never really pick a number? No.

[/ QUOTE ]

It is also clearly possible that you pick 7, even if P = 0. Thus, the idea of P=0 equaling "impossible" is hard to express using words. That was pretty much my point.

And don't worry -- I believe that you're right. There's no need to throw a hissy fit and offer to "play my game to prove me wrong". I simply wanted a little further explanation. Sorry to encroach upon your mathematical throne.

[/ QUOTE ]

I was offering you a chance to "plkay the game" to learn what the notion of 1/inf = 0 really means, not to show you up. You asked for layman's explanation, I tried to provide. Sorry I so erringly missed the boat.

And I commend you on your sense of humor the whole way. Your joy is infectious.

[Matt R.]

Oh my. Not sure if "his" in your post refers to me (probably), madnak, or the OP. I will assume it's me even though matheconomist was replying to madnak.

But, this is precisely the explanation I wanted. It is extremely intuitive. It should have been incredibly obvious this is the explanation I wanted based on my other posts. If you didn't realize this, you're very likely an idiot. Trained in math, sure. But most probably an idiot.

MathEconomist, thank you for your very well explained, non-condescending post. It is exactly what I was looking for -- you probably realized this immediately.

PS -- I'm quite sure tyrus didn't even think of this explanation until you posted it. Which is why he waited until after you posted it to say he thought it would be "over my head".

[Marko Schmarko]

It seems to me that this question is ridiculously poorly worded.

In science, a reasonable number of significant figures is always assumed (generally 3-4). To say "infinite sig figs" when referring to a necessarily measured quantity is somewhat nonsensical.

Also, I think bobman's subtle but insightful point makes the entire exercise invalid. Thermal energy is, indeed, quantized, and thus, has a discrete point at which one can be no more precise.

Thus, the probability that the temperature of a room is 72° (with infinite sig figs) is only zero if it happens to be physically impossible... which I suppose is pretty likely.

[Matt R.]

I know what 1/infinity means. Sorry to spoil your "everyone besides me is an idiot" fantasy world.

Did you actually read my other posts? It's pretty apparent what explanation I wanted. MathEconomist provided that. Or did you just see "omg he's questioning my excellence. He must not understand what inifinity means."

Your explanation missed the boat because you were explaining the wrong point.

And my "joke" wasn't intended as a funny funny ha ha joke. I meant it was a "joke" in the sense I didn't really believe the teacher was addicted to crack-cocaine, and that I disagreed with her point without fully understanding it. Sorry my stand-up fails to amuse you. Did you really believe I thought she was on drugs?

PS -- As I write this, I'm starting to realize you're trolling here. Maybe the joke is on me.

[tyrus72]

[ QUOTE ]
It seems to me that this question is ridiculously poorly worded.

In science, a reasonable number of significant figures is always assumed (generally 3-4). To say "infinite sig figs" when referring to a necessarily measured quantity is somewhat nonsensical.

Also, I think bobman's subtle but insightful point makes the entire exercise invalid. Thermal energy is, indeed, quantized, and thus, has a discrete point at which one can be no more precise.

Thus, the probability that the temperature of a room is 72° (with infinite sig figs) is only zero if it happens to be physically impossible... which I suppose is pretty likely.

[/ QUOTE ]

The context of the OP was a statistics teacher obviously making a point on continuous probability distributions. Neglecting this point dismisses the entire focus of the teacher's apparent lesson. "Infinite sig figs" is a way of specifying an exact value (versus an interval over the distribution) and thus should be read as such. This wasn't an engineering or physics class.

[Marko Schmarko]

Right. But, I think it's clear that her poor understanding of physics could only be caused by a long-standing, well-entrenched addiction to crack-cocaine.
-mark

[Leaky Eye]

The teacher should pick a better(correct) example to illustrate the concept.

Also, I used to sell crack to OP's teacher.

[DcifrThs]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
It is correct. Probability of being 72.000... is zero. Why? How many other possible temperatures are there? Infinity. What is 1/infinity? Zero.

[/ QUOTE ]

So for there to be a finite probability, you have to give a range rather than a point?



[/ QUOTE ]

Correct.

[/ QUOTE ]

the reason being that you can integrate a range in continuous space, which is what is causing the inifite number of possibilities. integrating a point yields zero b/c there is not enough area under the point to register.

EDIT: to be more clear, the range in continuous space to which i refer is representing sets of possible outcomes and their probability of being selected as one subset from all possible sets. any probability class really should begin with set theory and the point that any individual outcome of a random variable mapping an event to an outcome has a probability of zero. similarly, sums of disjoint outcomes are also zero.

i had a great deal of trouble initially accepting this and used to think of it this way: start with a temperature of 72degrees. now create a subset adding just above infinitely small increments to the set we want to look at (which starts off with a zero probability point at 72 degrees). now at SOME point, adding more and more of these possibilities to the set will give it a non zero probability. what helped me overcome this unacceptance was realizing that as the number of points added to 72 degrees reaches the threshhold of being assigned a positive probability, the subset therein approahces continuity asymptotically. without adding infinite # of these points, you can't make a truly continuous space, so the set's probability remains zero as a sum of disjoint possibilities.

Barron

[tyrus72]

[ QUOTE ]
But, this is precisely the explanation I wanted. It is extremely intuitive. It should have been incredibly obvious this is the explanation I wanted based on my other posts. If you didn't realize this, you're very likely an idiot. Trained in math, sure. But most probably an idiot.


[/ QUOTE ]

It doesn't seem to jive that the point of your posts is that this is all very non-intuitive and difficult to explain linguistically, but yet the "precise explanation" you were looking for is that the exact value is "measure zero" (a term the vast majority of people would not understand). But, whatever floats your boat.

[BeerMoney]

Going into ideas like quanta, etc, are really beside the point. The teacher is right, this is more of an issue of modelling. If I throw fourty die up in the air, and want to talk about the average of those die, we're going to model it as a continuous random variable. Its the only way to manage it.

In the case of the exact # having a prob. equal to 0, just look at it on a graph. When looking at a prob. for cont. rv's we look at areas over intervals. You have just one point, the area under one point is just a straight line. A straight line is 1 dimensional, while area is 2 dimensional. Therefore, there is no area.

Or, try assigning a probability to the chance of the temperature being 70. What would you like it to be?


Assume temperature is a uniform RV from 71.5-72.5

What is P(72<=X<72.1)? Ans .1
What is P(72<=X<72.01)? Ans .01
What is P(72<=X<72.001)? Ans .001
What is P(72<=X<72.0001)? Ans .0001
What is P(72<=X<72.00001)? Ans .00001

This interval just keeps getting smaller until you have one point..

Just look at it as a limit if that helps.

I think op and those "not getting it" need to perhaps give it a little more thought. Dcifrthis orginally said it bothered him, but he gave it more thought, and now it makes sense.

[PokerPadawan]

[ QUOTE ]
Isn't thermal energy quantized?

[/ QUOTE ]

While I think the OP's teacher's question was poorly posed, this is the most interesting response. Temperature is really just a way of saying how many possible states a system can be in, by way of its mean (per particle) energy. Each particle has a quantized state, but its energy is subject to the uncertainty principle. So you can imagine that a very large number (10^25, I guess, in a room) of air molecules, each with fuzzy energies, will make a very fuzzy average energy. Situations like these are part of the tricky boundary between classical physics and quantum physics. While it is probably true that an isolated system, no matter how complicated, will have energy states in integral units of the Planck energy or something, a room of air is not isolated and its state is time dependent. And since by the uncertainty principle it would take an infinite amount of time to measure the energy to infinite accuracy, you could never do this for a room of air.

This is probably a very confusing reply, but this how I would think about it. (I've almost finished a PhD in astrophysics.)

[Marko Schmarko]

I think you and Borodog should have an astrophysics phD duel.

Dissertation competitions.

[DcifrThs]

[ QUOTE ]
I think you and Borodog should have an astrophysics phD duel.

Dissertation competitions.

[/ QUOTE ]

yea i need help falling asleep /images/graemlins/wink.gif

Barron

[Borodog]

[ QUOTE ]
I think you and Borodog should have an astrophysics phD duel.

Dissertation competitions.

[/ QUOTE ]

He'd probably kick my ass. I've been out of the field for 3 years. That's like 21 dog and astrophysicist years.

[BeerMoney]

[ QUOTE ]
[ QUOTE ]
Isn't thermal energy quantized?

[/ QUOTE ]

While I think the OP's teacher's question was poorly posed, this is the most interesting response.

[/ QUOTE ]

If the teacher said "IF we model temperature as a continuous RV, what are the chances the temperature is exactly 72??" There is nothing wrong with this.

[LadyWrestler]

[ QUOTE ]
In a statistics class I am taking we discussed the probability of the temperature in a room being 72 degrees. Our teacher said that the probability of it being 72 degrees in the room was 0. This was because the temperature could always be measured more precisely to infinity (78.0000…1). This seems impossible because if you assume that the temperature drops from 73 degrees to 71 degrees then while its dropping it must pass through exactly 72 degrees not (78.00000…1). So since there is a number that is exactly 72 degrees then there must at least some chance (0.0000…1) that the temperature is exactly 72 degrees in the room. After much heated discussion she told us that the probability of it being 72 degrees was 0, not a very small number close to 0. Is this correct or not?

[/ QUOTE ]

Not.

[PokerPadawan]

No, of course not. That is exactly what the teacher should've said.

[LadyWrestler]

In the real world this is crap, eloquent crap but still crap. The answer to the original poster's question is still "not" and will always be "not". Your post is fun to read though. You should run for office! /images/graemlins/smile.gif

Have a great day!

[Bork]

[ QUOTE ]
It is correct. Probability of being 72.000... is zero. Why? How many other possible temperatures are there? Infinity. What is 1/infinity? Zero.

[/ QUOTE ]

Nope.

1 divided by infinity does not equal zero

infinity is a concept not a number, as such you cannot use it with mathematical operators. If you mean the limit of 1/x as X approaches infinity then yes the answer is zero. What do you think the correct way to analyze 1/red or 1/soul is? They are nonsensical just like 1/infinity

Imagine space is continuous. Take a cubical chunk of space and divide it into infinitely many spaces, does the volume of the cube become zero because each sub-space has a volume of zero? No, because each sub-space has some non-zero volume.
The same can be said of a solid mass. Assume that it is possible to split it into infinitely many pieces, obviously each has some mass greater than zero.


Your teacher is wrong. The probability is a very small number not 0. (if we assume tempurature is continuous)

[PokerPadawan]

Please, people. The teacher was trying to make a point about continuous probability distributions. They are defined mathematically such that the probability of any particular value of a random variable X is 0 (P(x=X) = 0). Instead, they have finite probability over an infinitesimal range in the random variable (P(X < x < X+dX) != 0).

That is the only way you could manipulate them through calculus. If they were not like this, you could prove through Riemannian sums that the distribution would sum to infinity.

[PokerPadawan]

[ QUOTE ]
[ QUOTE ]
I think you and Borodog should have an astrophysics phD duel.

Dissertation competitions.

[/ QUOTE ]

He'd probably kick my ass. I've been out of the field for 3 years. That's like 21 dog and astrophysicist years.

[/ QUOTE ]

The maxim "if you don't use it, you lose it" is very true in this field. I feel like I've forgotten a lot just since I was an undergrad.

[Bork]

If the mathematical definition says the probability is zero then the mathematical definition is wrong. Actually the definition can't technically be wrong, but the application to anything beyond math is going to result in falsehoods.

Calculus is not some mighty truth teller. Simply saying "calculus tells us X so X", is not a satisfactory answer. Calculus is one model, and like any model can get incorrect results if applied to the wrong problems in the wrong ways.

If the teacher is making a point about math then fine. If he/she is trying to illustrate a point about the real world then he/she is mistaken. (even if we presuppose continuous temps in our 'real world')

[Marko Schmarko]

Bork: what would the probability be?
- mark

[Bork]

[ QUOTE ]
Bork: what would the probability be?
- mark

[/ QUOTE ]

I already answered this in my first post in this thread. In that post I made an argument for the conclusion that it is not zero.

Here is the argument again:
Imagine space is continuous. Take a cubical chunk of space and divide it into infinitely many spaces, does the volume of the cube become zero because each sub-space has a volume of zero? No, because each sub-space has some non-zero volume.
(that isn't a formal argument, but the idea is if each subspace had zero volume then the total of the subspaces must be 0, but if each has some infinitely small volume then taking all of them together will result in the true volume of the cube)

It is a very small non-zero number. Don't know exactly what it is but I do know it isn't zero and that it is low. I can answer conceptually and just say it is infinitesimally small.

Imagine an apple is continous, does it follow from the apple being divided infinitely many times that each piece has zero mass? If every piece of an object has zero mass then the object must have zero mass itself, no? The apple doesn't have zero mass so obviously each infinitesimally small piece has some infinitesimally small mass.

[madnak]

It doesn't follow that if every piece has zero mass, so does the whole apple.

You are using the assumption that x*0=0. I posted a similar argument earlier. But apparently infinity*0 doesn't = 0. It seems a bit strange to me, but there you have it.

[Bork]

I am not saying anything about x*0, where x is infinity. I already said we ought not use mathematical operators with infinity because it is a concept.

Just do a quick thought experiment where you think about the problem in conceptual and logical terms and it should become clear that the right answer can't be zero.

Odd that all these math people think you can use infinity with mathematical operators, I asked two math profs today and they both said that you can't and they only do that because it is convenient when talking about limits and limit like stuff.

http://mathforum.org/library/drmath/view/62486.html
Apparently Dr. Math agrees. (although his site looks like it's aimed at mere k-12 I think he got this right) /images/graemlins/crazy.gif

also: http://en.wikipedia.org/wiki/Infinity
makes it pretty clear in the first line that infinity is a concept, though later it says there is some disagreement about this, I would point out that doesn't mean infinity is a number just that some people think it is.

[madnak]

Dr. Math is wrong. The conclusion reached on the 0.999... thread was that 0.999...=1. Zeno's paradox is similar. If 0.5 + 0.25 + 0.125... were not equal to 1, then space and time could not exist.

[Bork]

I don't see where he says that .9 repeating doesnt equal one. (he doesn't in the link I posted) Further I don't see how it is relevant.
In fact he says this.

http://mathforum.org/dr.math/faq/faq.0.9999.html


Just talking about .999... as equaling one is odd because the series of nines goes on for infinity. So .999... is parasitic on the concept infinity (ie one of its parts is the concept infinity) so it shouldn't be analyzed as though it is a numerical value. Alternatively .9 repeating is just another way of refering to the numerical value 1. I suspect the people in that thread were using it in the second way. (i.e. the limit of .9+ .09 .....)

[madnak]

And in the OP, the teacher is talking about the limit of 0.1*0.1*0.1... which evaluates to 0. Thus the answer is 0.

[Matt R.]

I'm going to sort of go with Bork on this one. Especially since tyrus formally proved to me that he is an idiot.

I think the fact that you get a zero probability from the OP's question is a consequence of the definitions/axioms laid forth in Calculus and probability theory. As someone else pointed out, it is so you can evaluate the probabilities using integrals. However, logically, the probability cannot equal zero, as that directly implies impossibility. It clearly cannot be impossible as has been shown numerous times.

Basically P=0 is correct in terms of the definitions laid forth for continuous random variables. But those same definitions are slightly wrong when applied to certain real world problems. They still are useful of course, but they yield non-sensical answers sometimes when you say things like the probability of picking an integer 7 is zero (because there is an infinite number of integers). Thank you tyrus for giving me that example. Can we be friends? Because I sure do love you. /images/graemlins/heart.gif

Matt

[madnak]

How does it directly imply impossibility?

[bobman0330]

[ QUOTE ]
Basically P=0 is correct in terms of the definitions laid forth for continuous random variables. But those same definitions are slightly wrong when applied to certain real world problems.

[/ QUOTE ]

The theory of continuous random variables is, strictly speaking, inapplicable to real-world problems. Whether or not P = 0 implies physical impossibility isn't particularly important, because there are no true CRVs in real life.

[Matt R.]

The probability of an event occuring is 0 <= P(E) <= 1. 0 means the event never occurs (thus it is impossible) and 1 means it occurs every time. This is from my understanding, but I think it follows directly from the axioms of probability theory. I'm fairly certain it would be easy to construct a proof that says when P=0 an event is impossible. But, I also think it is the very definition of the probability being equal to zero, so it may be taken as true without a proof.

Anyway, another way to look at it is if an event occurs with probability equal to one it is the only event that occurs in that sample space. Since probabilities sum to 1 then no other event in that sample space can occur, and the sum of the probabilities of any other event in that sample space must be 0.

Oh, I'm also now convinced that 1/infinity does not equal zero. Now that I think about it, every time I've considered such a term it has been in calculus when evaluating a limit. Thanks for pointing that out Bork, 1/infinity when not in terms of limits doesn't mean anything.

I'm open-minded though, so if someone more knowledgeable in math (that is not a troll) disagrees I'd reconsider.

[madnak]

I would like to see that constructed from the axioms. I'm not going tobelieve that without proof.

I may not know math, but this constant contradiction is absurd. If 0.999...=1, then 1/infinity=0. So somebody is clearly wrong.

[Matt R.]

The .999... = 1 proofs were taken in terms of limits. Saying "1/infinity = 0" is non-sensical. When you say "1/x = 0" as x approaches infinity, then you're talking in terms of limits. You're not really dividing by infinity, you're simply saying as x becomes larger and larger 1/x = 0. There's a subtle difference there.

I'm no mathematician though, and my math is pretty sloppy still, so I wouldn't have caught that had Bork not pointed it out.

As far as saying P=0 implies impossibility, I'm looking at my probability theory textbook right now (haven't had it in awhile... so excuse my ignorance, or at least lack of confidence, when explaining this stuff), and a quote from the section "axioms of probability" says, "One way of defining the probability of an event is in terms of its relative frequency." Axiom 1 is that 0 <= P(E) <= 1 where E is an event in the sample space S. If the frequency of an event occuring is zero, then it never occurs by definition. If something never occurs, it is impossible for it to occur.

I'm pretty sure what you're asking IS an axiom. Not sure... but pretty sure. If not, I'm not going to attempt to construct a formal proof. If there are any real mathematicians in here, I may make them angry.

[Matt R.]

OK, I just realized saying "something never occurs" does not really equate to "it is impossible for it to occur". I think this may be where you're in disagreement.

If a formal proof were to be constructed, I think one way of going about it may be like this (I will not attempt to use formal language, just logic)

From axiom 1 we saw that probabilities are 0 <= P(E) <= 1. Axiom 2 states that P(S)=1 which simply means that with probability 1 the outcome will be in sample space S. Axiom 3 (I can't type the symbols here) states that "for any sequence of mutually exclusive events the probability of at least one of these events occurring is just the sum of their respective probabilities".

From axiom 2, we see that the outcome must be a point in the sample space. If you take another event E2, and say that it occurs with probability one, then from axioms 2 and 3 the sum of every other probability has to be zero. Thus it is impossible for any other event in the sample space to occur. Think of it as event E2 "taking up" all the space in S, so it's impossible for any other event to occur there. This is why when you say P=0 it implies impossibility -- only possibilities with a non-zero probability can occur.

I'm pretty sure that's a good explanation for the logic behind it. Definitely ugly and not formal though.

[madnak]

Okay, but an infinite sample space screws that up. And it's not "never occurs" either. What proportion of rational numbers include the digit 3 in their decimal representation?

If .999...=1, then 1-.999...=0. A situation like that in the OP can be described in these terms. Take a starting field of 1. Now subtract the chance that n<72 or n>=73. Then the chance that n>=72.1 Then the chance n>=72.01.

What I mean is, the probability that 72 is not the temperature is 0.9 + 0.09 + 0.009... That is to say, 0.999...

If 0.999...=1, then the probability that the temperature isn't 72 is 1. By your reasoning above, this implies that the probability of 72 being the temperature is 0.

In order for the answer to be something other than 0, .999... cannot equal 1.

[Matt R.]

madnak,
I had a frustratingly long reply typed up, but I'm getting tired, and I think my logic was very circular. I'll have to think about this a bit, but I need to hit the sack right now.

[MathEconomist]

Please just stop, this isn't going to go anywhere. In order for probability theory to work at all, continuous distributions must assign probability zero to finite collections of points. There is no doubt about this. Whether or not this "is applicable to real life" isn't really relevant, since the applications of probability theory to real life inference problems are not in doubt.

If you really want to understand the deepest reasons why this works out this way, you can take a course in measure theory and then a graduate level probability theory course that uses the measure theory. For just about everyone in the world, it's not important to understand it that deeply and you should just accept that this is part of the definition of a continuous distribution.

[PokerPadawan]

Thank you, M. E.

[Piers]

You’re using a bad model, leading to stupid questions.

Its not that the probability of temperature being 72 is zero, its that using a model where it is meaningful to ask what the temperature of a single point is, is not useful.

Instead ask about ranges of temperatures.

[Bork]

OP
[ QUOTE ]
In a statistics class I am taking we discussed the probability of the temperature in a room being 72 degrees. Our teacher said that the probability of it being 72 degrees in the room was 0.

[/ QUOTE ]

This is sounds suspiciously like the teacher is taking a stipulated math definition and using it to make claims about the way thinks would actually be (if thetemperature were continuous).

[ QUOTE ]

It's not important to understand it that deeply and you should just accept that this is part of the definition of a continuous distribution

[/ QUOTE ]
I am accepting it is a consequence of some definitions which are useful to mathematicians. If they didn't define it as zero they would get some messed up results. However this doesn't mean it actually would be zero in the case described by the teacher. I offered arguments which I think prove that it is not zero which have been ignored.

Seriously folks, which are you more confident in, some axioms that probability theory just stipulates to make numbers add up, or a simple thought experiment?

[Bork]

[ QUOTE ]
And in the OP, the teacher is talking about the limit of 0.1*0.1*0.1... which evaluates to 0. Thus the answer is 0.

[/ QUOTE ]

The teacher does make reference to limits and this explains the teachers erroneous results, if the teacher is making a claim about the way things would be in a room with continuous temps. If the teacher is simply making a math claim in a dumb way then I will take the forums word for it that the math is right. Honestly though why would it need to be posted if it is a math question? Its pretty clear the teacher made it sound like thats the way things would actually be, not just the way they are in math world. If the teacher had said well thats what math says, so thats they way things are in math world then the OP would not have asked the question.

[madnak]

(Disclaimer: I don't know any math at all past basic algebra)

I think it is relevant in many cases. For example, there are an infinite number of sets of basic assumptions about the world.

Therefore no set of assumptions can be mathematically justified. They are all mathematically worthless. It follows that our mechanism for selecting basic assumptions is not probabilistic. So we must use some other mechanism (contact with God, parsimony, etc).

I do wonder about one things, however. If the "total" probability must equal 1, then P=0 for every option doesn't make much sense. After all, 0+0+0...=0. So the sum of all probabilities is 0. My interpretation is that this represents the basic nature of the question. You can't create space out of no-space. The entire fundamental point is as simple as, a point has no area. P is a way to look at an "area" of possibility, but in order to have area you have to have length ("range").

No matter how many points you have, you can't make a line (or any polygon or polyhedron). A square isn't "made of points" in any sense. ALL probability involves ranges. With a finite number of options those ranges are set, so we see the specific options as "points." But they are not points, they are ranges. When we roll a die we have "0-1, 1-2, 2-3, 3-4, 4-5, 5-6" as our options. You can't roll 4.3 on a die, but if we represent the system graphically that is what it looks like. Anything between 0 and 1 is a 1, anything between 1 and 2 is a 2, etc... This is how a computer program needs to determine die rolls.

So when you say what is the probability of something, you are asking what area of the total "1" is taken up by that case. Since each case in this kind of situation represents a "point" it has no area. The area of the specific case is definitely 0. And that is that. There really is no likelihood that a specific event will occur. Mathematically, the specific event that does occur is arbitrary. Only ranges are relevant.

And that is a mathematical way to express the sort of metaphysical ontological paradox. You can't talk about the "likelihood of God." It doesn't exist. There are an infinite number of Gods and an infinite number of no-Gods. Even with 1-to-1 correspondence I don't think you can speak of any "likelihood."

[LadyWrestler]

..."if you assume that the temperature drops from 73 degrees to 71 degrees then while its dropping it must pass through exactly 72 degrees"...
---

Exactly.

Or...If you and I started wrestling on the floor in that room, the temperature would quickly reverse course and pass through exactly 72 on the way up to well over 100...

Smokin' hot!

Case closed. /images/graemlins/laugh.gif

[Prodigy54321]

[ QUOTE ]
..."if you assume that the temperature drops from 73 degrees to 71 degrees then while its dropping it must pass through exactly 72 degrees"...
---

Exactly.

Or...If you and I started wrestling on the floor in that room, the temperature would quickly reverse course and pass through exactly 72 on the way up to well over 100...

Smokin' hot!

Case closed. /images/graemlins/laugh.gif

[/ QUOTE ]

that made me laugh, well done

[Matt R.]

[ QUOTE ]
Please just stop, this isn't going to go anywhere.

[/ QUOTE ]

I appreciate your input earlier, but all I'm really trying to do is gather a full understanding of this. Does this annoy you or something?

madnak's last post last made me realize one of my assumptions was wrong. I believe madnak is trying to understand it as well. There's no need to respond to tell us to stop simply because our lack of knowledge in measure theory isn't up to par. I'm probably not going to take the course, but I still want to think about it.

I think I have a full understanding now (minus the terminology) of why you state the probability is zero in regards to the question. So the debate clearly did some good.

Anyway, thanks again for your first post in the thread -- my understanding kind of looped back around to your point after I thought about it some more.

[BrickTamlin]

Those who agree with Bork, MattR, and Lady Wrestler are absolutley and totally wrong. Enjoy your ignorance, however.

[Matt R.]

Thank you BrickTamlin, post count one.

I hope your future posts are just as enlightening. I'm glad our debate was interesting enough to cause you to create an account and put us in our place. Good argument, by the way.

[BrickTamlin]

Your arguments speak loudly enough for themselves.

[Matt R.]

[ QUOTE ]
Your arguments speak loudly enough for themselves.

[/ QUOTE ]

Spoken like someone with a true understanding of the material.

BrickTamlin: "I disagree with the guy who has stated repeatedly throughout the thread that he's never studied this stuff!"

Bravo, sir. Or did you "overlook" the parts where I said I'm trying to gain an understanding of this myself?

[madnak]

[ QUOTE ]
I believe madnak is trying to understand it as well.

[/ QUOTE ]

This is correct. I think we both have a better grasp on it now.

[BrickTamlin]

[ QUOTE ]
[ QUOTE ]
Your arguments speak loudly enough for themselves.

[/ QUOTE ]

Spoken like someone with a true understanding of the material.

BrickTamlin: "I disagree with the guy who has stated repeatedly throughout the thread that he's never studied this stuff!"

Bravo, sir. Or did you "overlook" the parts where I said I'm trying to gain an understanding of this myself?

[/ QUOTE ]

You obviously aren't trying to gain an understanding, but rather trying to refute those trying to provide you that understanding. And for that, you deserve to wallow in your ignorance.

[Matt R.]

OK, this is my last reply to you, as I'm not going to get snookered by another troll in this thread.

Constructing potential counterarguments around the finer points that you don't have a full grasp on is how learning takes place. I'm sure you've never asked, "why is x true, given y" when confronted with a non-intuitive problem. Want to know why I'm sure? You clearly have the creativity of a doorknob and have never constructed an argument of your own in your life.

Keep reading those textbooks and believing everything you're told. That way you can come on an internet message board and let me know that my argument doesn't jive according to page 12 line 43 of "Intro to Probability Theory for Internet Douchebags".

[Bork]

[ QUOTE ]
I do wonder about one things, however. If the "total" probability must equal 1, then P=0 for every option doesn't make much sense. After all, 0+0+0...=0. So the sum of all probabilities is 0. My interpretation is that this represents the basic nature of the question. You can't create space out of no-space. The entire fundamental point is as simple as, a point has no area. P is a way to look at an "area" of possibility, but in order to have area you have to have length ("range").

No matter how many points you have, you can't make a line (or any polygon or polyhedron). A square isn't "made of points" in any sense. ALL probability involves ranges. With a finite number of options those ranges are set, so we see the specific options as "points." But they are not points, they are ranges. When we roll a die we have "0-1, 1-2, 2-3, 3-4, 4-5, 5-6" as our options. You can't roll 4.3 on a die, but if we represent the system graphically that is what it looks like. Anything between 0 and 1 is a 1, anything between 1 and 2 is a 2, etc... This is how a computer program needs to determine die rolls.

So when you say what is the probability of something, you are asking what area of the total "1" is taken up by that case. Since each case in this kind of situation represents a "point" it has no area. The area of the specific case is definitely 0. And that is that. There really is no likelihood that a specific event will occur. Mathematically, the specific event that does occur is arbitrary. Only ranges are relevant.

[/ QUOTE ]

This is circular. When I divided up the cube in my head I divided it into infinitesimally small subspaces. When you call them points you are presupposing that they have zero volume. What they actually are is the smallest possible(non-zero volume) space. This is analogous to the continous temp where each unique temperature has the smallest possible non-zero probability.

The whole point (no pun intended) is that if have a finite space which has volume it cannot consist totally of parts which each have zero volume. So the smallest possible parts are not points.

[madnak]

BrickTamlin -> show all user's posts.

He won't last long.

[madnak]

There are no "smallest possible parts." No matter how small you get, it can always be further subdivided. A finite thing can't be subdivided into infinite "pieces." The sum of the pieces can never add up to the actual thing.

[Bork]

[ QUOTE ]
There are no "smallest possible parts." No matter how small you get, it can always be further subdivided.

[/ QUOTE ]


Dividing them further can never make them have a volume of zero, right?

[ QUOTE ]
A finite thing can't be subdivided into infinite "pieces." The sum of the pieces can never add up to the actual thing.

[/ QUOTE ]

I think, the 'sum' of the pieces could add up the total volume because in crude terms there are very many of them and they are very small. Where is the problem? The problem isn't a logical one. The problems that have been presented so far are in terms of using infinity as an actual number rather than a concept.

[Matt R.]

[ QUOTE ]
BrickTamlin -> show all user's posts.

He won't last long.

[/ QUOTE ]

Yeah... I don't know why I let guys like that get to me on internet forums. /images/graemlins/blush.gif

One of my biggest pet peeves is closed-mindedness surrounding the learning process and academics. e.g. people telling you "no this is wrong" but they can't tell you why or can't tell you where your arguments fail. Some people did a great job of providing logical arguments in this thread... others not so well.

I need to start letting idiotic comments roll off my back, but it's hard to do when they're directed right at you. At least I learned something out of this /images/graemlins/grin.gif.

[BrickTamlin]

[ QUOTE ]

One of my biggest pet peeves is closed-mindedness

[/ QUOTE ]

Irony is priceless.

[madnak]

Yeah, it can be tempting. But they won't go away unless you ignore them.

[BrickTamlin]

[ QUOTE ]
Yeah, it can be tempting. But they won't go away unless you ignore them.

[/ QUOTE ]

And this from the math genius who brought us:

[ QUOTE ]

Excuse my algebraic math, but some things just don't seem right here. i is infinity

1/i = 0

1 = 0i

Huh?


[/ QUOTE ]

[Marko Schmarko]

Brick:
You forgot to keep reading his post.
[ QUOTE ]
e.g. people telling you "no this is wrong" but they can't tell you why or can't tell you where your arguments fail.

[/ QUOTE ]
.. that was right after the part you quoted. I would like to see concise, lucid counter-arguments.

This seems like it may be a disagreement between statistical theory and practical reality.

[BrickTamlin]

[ QUOTE ]
Brick:
You forgot to keep reading his post.
[ QUOTE ]
e.g. people telling you "no this is wrong" but they can't tell you why or can't tell you where your arguments fail.

[/ QUOTE ]
.. that was right after the part you quoted. I would like to see concise, lucid counter-arguments.

This seems like it may be a disagreement between statistical theory and practical reality.

[/ QUOTE ]

He dismisses math explanations for his question but then follows that dismissal in later posts with statements suggesting "1/inf=0 implies that 0*inf=1, and thus 1/inf cannot equal 0". He thus has shown that he doesn't have the capacity to understand any explanations or the unwillingness to concede that his math skills are limited and that he should think more about what people are telling him. Either case, discussing it further with him is futile. But mocking him is fair game.

[Leaky Eye]

Welcome to ignore, douche.

[BrickTamlin]

[ QUOTE ]
Welcome to ignore, douche.

[/ QUOTE ]

uh huh.