the probablity of temperature

[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?

[BruceZ]

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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?

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It's not because temperature, being average kinetic energy, is quantized into a finite number of discreet levels according to quantum mechanics.

If you were to model it as a continuous process, then the probability of it being any particular value would be zero, even though it can take on a range of values. You can also think of it as an infinitesimal quantity f(t)dt, where f(t) is the probability distribution which must be integrated over a range to give a probability. The real numbers simply provide a convenient mathematical model for reality. Not all conclusions obtained from it are supported by physics, as this example demonstrates.

[RiverTheNuts]

It will and it wont be... over some range of temperatures, 72.000... exists, but because there are infinitely many numbers between 71.999 and 72.0001 it will never be exactly 72 degrees, which is why you must take integrals over a range. The integral of a function from x to x is always zero, even though a small range including x would have a positive value. In discreet probability you can either have 0, 1, 2, or 3, whereas in continuous statistics there are infinite numbers in the set of [0,3], therefore, assigning any of them a positive probability value would be wrong, because any number * infinity > 100%

[BruceZ]

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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?

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It's not because temperature, being average kinetic energy, is quantized into a finite number of discreet levels according to quantum mechanics.

If you were to model it as a continuous process, then the probability of it being any particular value would be zero, even though it can take on a range of values. You can also think of it as an infinitesimal quantity f(t)dt, where f(t) is the probability distribution which must be integrated over a range to give a probability. The real numbers simply provide a convenient mathematical model for reality. Not all conclusions obtained from it are supported by physics, as this example demonstrates.

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I should further point out that IF you model temperature as a continuous real valued function, then your assertion that it must take on a particular value of 72 at some instant in time while changing from 73 to 71 is absolutely correct, and this is an important theorem from real analysis called the Intermediate Value Theorem. The same is true of the velocity of a vehicle changing from one value to another under the Mean Value Theorem.

It remains true that the probability of being equal to 72 at any given time is still zero (or infinitesimal) under this model, while the probability that it was equal to 72 at some time during the interval over which it was changing would be 1. When dealing with an infinite number of possible outcomes, a probability of zero does not necessarily mean that an event can't happen, nor does a probability of 1 mean that it is certain to happen. Here we have an example of an event with a probability that is zero for every time instant, yet it is certain to happen at some time instant! This is a consequence of the model, not necessarily of reality.

[vhawk01]

You should really tell your teacher to use a different example. Tell her to pick something that actually CAN be measured to an infinitely diminishing accuracy. Weight, temperature, etc. are not good examples. They are a collection of quanta, and therefore have some actual finite accuracy. It would be like saying you can never know the exact number of people on earth, because you can measure to the .0000001 of people. Nonsense.

[drbst]

Nonsense, indeed.

The teacher wanted to explain the fact that on continuos scales the probability mass on every finite set of point is zero. This theoretical result she tried to illustrate with an entity that is measured on a continous scale, the temperature. She could also have chosen the height, weight, or age of the students, it would be all the same and in real life all examples would have been equally wrong.

I remember that I argued once in the same way during my studies and the reply I got my professor went like this: "The fact that in real life every existing scale is discrete in the end has no effect on the validity of the theoretical model. You must learn to think in the framework of the model."

If I were your teacher I would in the future skip all attempts to explain the models by examples and let the theory and its beauty speak for itself. /images/graemlins/wink.gif

[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?

[alThor]

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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?

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It's not because temperature, being average kinetic energy, is quantized into a finite number of discreet levels according to quantum mechanics.

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Of course this is not yet a proof; you forgot to show that 72.000000... just happens to be exactly one of those levels. (What were the odds of that?!) /images/graemlins/smile.gif /images/graemlins/smile.gif

[BruceZ]

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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?

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It's not because temperature, being average kinetic energy, is quantized into a finite number of discreet levels according to quantum mechanics.

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Of course this is not yet a proof; you forgot to show that 72.000000... just happens to be exactly one of those levels. (What were the odds of that?!) /images/graemlins/smile.gif /images/graemlins/smile.gif

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Oh, yeah. Good point. The probabilty is almost certainly 0. Almost certainly 0? /images/graemlins/grin.gif

[AcesOffsuit]

Much could be said here about the density of the real numbers and what it really means to be continuous on an interval.

But it wouldn't matter much, now would it did?

One of my old professors used to say, "Math is hard, let's go shopping." I tend to agree.

So instead let me tell you of the time I had pocket blah blah blah...