New Paradox: Zero Probability But Not Impossible?!

[Kharakov]

Assuming 2x4s are 2x4 will screw you up though, since they are ~1.5 x 3.5.

 

[Alcoholic Actuary]

Events with a probability of zero happen all the time. Once you are comfortable with probability as a measurement, then you should be comfortable with the idea that 1/(Extremely Large Number) = 0 = Extremely Unlikely (but not impossible). OTOH, things that are impossible, we assign a probability of zero.

If I have a six sided die, whose faces are numbered 1 to 6, then the probability I roll a 7 is zero - because it is impossible. There is no 7 on the die. The probability I roll 10000 consecutive 6's is also zero, but not impossible - just extremely unlikely (but no more unlikely than any particular ordered set of 10000 outcomes).

aa

That's a fairly loose definition of zero. It's just calling a non-zero number zero because it's pretty close to zero.

..which we do all the time, when we're taking measurements of things. A 2 x 4 is not precisely 2" by 4" down to the angstrom. Even the device used to measure has uncertainty in it.

aa

And that's fine. When working with 2x4's, you don't need any kind of precision greater than "close enough". That doesn't mean that the same standard occurs with everything and especially not with mathematical proofs.

Precisely. So we're going to have to pick the mathematical scenario and then append the level of precision. If we are dealing with the purely theoretical, then the proof is even easier. As another poster pointed out we define a continuous probability distribution and ask for the probability that X=some specific point on the distribution. It is exactly 0 but clearly not impossible since it is within the distribution. qed.

The problem is that theoretical probability distributions almost have a desire for some application. So folks walk this tightrope back and forth between declaring some probability non-zero even after their calculation of some ridiculously small number outweighs it's precision by multiple orders of magnitude. It's a real world application whose probability calculation requires highly theoretical assumptions that fall apart when tested back in the real world. What is the probability I roll 10000 consecutive 6's on a die? If you say it's 1/(6^10000) and that's materially different from zero, I will take issue with it. Theoretically, it might be calculated that way - assuming a perfectly fair die (which does not exist in reality), a perfectly controlled experiment that can randomize the starting position of the die and the force and angle of the throw (which does not exist in reality), etc. Any of a small subset of initial conditions during the course of the experiment could significantly alter the calculation of the outcome of 10000 rolls of the die. Finally the actual magnitude of the result is just not worth knowing back in the real world. For all intents and purposes the probability of the event is zero. General non-zero measurements of probability may not be wrong, but any one specific non-zero estimate is almost certainly not right and ought not alter our behavior or decision-making process derived from one that treats the measurement as zero.

The paradox in the OP presents darts on a dartboard as some sort of paradox. It isn't. In the real world we measure probabilities as zero for events all the time. The dartboard is a valid example. If it is meant as a theoretical example, then I can envision a dartboard with an area consisting of infinite points and the dart as one of the points (same as integrating a continuous distribution at a single point on the distribution). If it is meant to be a real world example with a real dartboard and a real dart, then I am comfortable in relaxing the assumptions around the theoretical probability distribution and reporting my probability measurement to, say 5 significant digits, and calculating the probability as 0.00000.

aa

 

[Alcoholic Actuary]

Events with a probability of zero happen all the time. Once you are comfortable with probability as a measurement, then you should be comfortable with the idea that 1/(Extremely Large Number) = 0 = Extremely Unlikely (but not impossible). OTOH, things that are impossible, we assign a probability of zero.

If I have a six sided die, whose faces are numbered 1 to 6, then the probability I roll a 7 is zero - because it is impossible. There is no 7 on the die. The probability I roll 10000 consecutive 6's is also zero, but not impossible - just extremely unlikely (but no more unlikely than any particular ordered set of 10000 outcomes).

aa

That's a fairly loose definition of zero. It's just calling a non-zero number zero because it's pretty close to zero.

..which we do all the time, when we're taking measurements of things. A 2 x 4 is not precisely 2" by 4" down to the angstrom. Even the device used to measure has uncertainty in it.

aa

How about a dartboard as continuous as the real number line; could we then discern the difference between the "possible" 0 and the "impossible" 0?

Sorry I missed this one earlier.

Are you saying that the probability distribution is defined over the real number line? Then all possible 0's are exact outcomes on the real number line. An example of an impossible outcome would be the probability is equal to the square root of -1. Imaginary numbers would not be defined in the distribution and so would be assigned a value of 0.

aa

 

[ryan]

Events with a probability of zero happen all the time. Once you are comfortable with probability as a measurement, then you should be comfortable with the idea that 1/(Extremely Large Number) = 0 = Extremely Unlikely (but not impossible). OTOH, things that are impossible, we assign a probability of zero.

If I have a six sided die, whose faces are numbered 1 to 6, then the probability I roll a 7 is zero - because it is impossible. There is no 7 on the die. The probability I roll 10000 consecutive 6's is also zero, but not impossible - just extremely unlikely (but no more unlikely than any particular ordered set of 10000 outcomes).

aa

That's a fairly loose definition of zero. It's just calling a non-zero number zero because it's pretty close to zero.

..which we do all the time, when we're taking measurements of things. A 2 x 4 is not precisely 2" by 4" down to the angstrom. Even the device used to measure has uncertainty in it.

aa

How about a dartboard as continuous as the real number line; could we then discern the difference between the "possible" 0 and the "impossible" 0?

Sorry I missed this one earlier.

Are you saying that the probability distribution is defined over the real number line? Then all possible 0's are exact outcomes on the real number line. An example of an impossible outcome would be the probability is equal to the square root of -1. Imaginary numbers would not be defined in the distribution and so would be assigned a value of 0.

aa

So does an "impossible probability of zero" equate to an "almost impossible probability of zero"? Or, theoretically, is there an infinitesimally small probability?

 

[Juma]

So does an "impossible probability of zero" equate to an "almost impossible probability of zero"? Or, theoretically, is there an infinitesimally small probability?

There is only one infinitisemal small number: 0.

 

[Alcoholic Actuary]

Events with a probability of zero happen all the time. Once you are comfortable with probability as a measurement, then you should be comfortable with the idea that 1/(Extremely Large Number) = 0 = Extremely Unlikely (but not impossible). OTOH, things that are impossible, we assign a probability of zero.

If I have a six sided die, whose faces are numbered 1 to 6, then the probability I roll a 7 is zero - because it is impossible. There is no 7 on the die. The probability I roll 10000 consecutive 6's is also zero, but not impossible - just extremely unlikely (but no more unlikely than any particular ordered set of 10000 outcomes).

aa

That's a fairly loose definition of zero. It's just calling a non-zero number zero because it's pretty close to zero.

..which we do all the time, when we're taking measurements of things. A 2 x 4 is not precisely 2" by 4" down to the angstrom. Even the device used to measure has uncertainty in it.

aa

How about a dartboard as continuous as the real number line; could we then discern the difference between the "possible" 0 and the "impossible" 0?

Sorry I missed this one earlier.

Are you saying that the probability distribution is defined over the real number line? Then all possible 0's are exact outcomes on the real number line. An example of an impossible outcome would be the probability is equal to the square root of -1. Imaginary numbers would not be defined in the distribution and so would be assigned a value of 0.

aa

So does an "impossible probability of zero" equate to an "almost impossible probability of zero"? Or, theoretically, is there an infinitesimally small probability?

No. It is a special definition. Through the construction of the experiment, we first determine what is possible and what is impossible. All things impossible we assign a probability of zero - they occupy none of the probability distribution's sample space. If our theoretical example is best defined using an exponential distribution with a mean of 2, then the probability that X = -1 is equal to zero by definition, because -1 is impossible. The exponential distribution is only defined for values of x >= 0. However, the probability that X = 2 (the mean of the distribution) is also equal to zero - but obviously possible.

Logically, if something is impossible, then it's probability is 0, is always true (as is the contrapositive). The converse - If the probability is zero, then it is impossible - and the inverse - if it is possible, then it's probability is not 0 - are not necessarily true.

aa

 

[Speakpigeon]

That's a fairly loose definition of zero. It's just calling a non-zero number zero because it's pretty close to zero.

..which we do all the time, when we're taking measurements of things. A 2 x 4 is not precisely 2" by 4" down to the angstrom. Even the device used to measure has uncertainty in it.

aa

I'm not seeing how that is relevant to the logic forum. The question clearly implies mathematically ideal objects.

In an ideal world we would stick to the point but we're not in one. Or maybe we are?

It's also relevant to point at the similarities of language and the traps that involves.

Also, the OP seems to implies a purely logical question only relevant to an ideal world yet it seems clear to me that ryan is mixing in a fair amount of realistic possibilities, not just in the OP but in subsequent posts, showing I think that he hopes to be able to deduce something about the material world from his paradox. The OP is about our representation(s) of reality I think.
EB

 

[Speakpigeon]

Assuming 2x4s are 2x4 will screw you up though, since they are ~1.5 x 3.5.

Which in an ideal world would be 3,81 x 8,89... centimeters.

That can screw you up too if you go to Mars.
EB

 

[ryan]

So does an "impossible probability of zero" equate to an "almost impossible probability of zero"? Or, theoretically, is there an infinitesimally small probability?

There is only one infinitisemal small number: 0.

Don't you mean that there is only one infinitesimally small real number 0?

 

[ryan]

So does an "impossible probability of zero" equate to an "almost impossible probability of zero"? Or, theoretically, is there an infinitesimally small probability?

No. It is a special definition. Through the construction of the experiment, we first determine what is possible and what is impossible. All things impossible we assign a probability of zero - they occupy none of the probability distribution's sample space. If our theoretical example is best defined using an exponential distribution with a mean of 2, then the probability that X = -1 is equal to zero by definition, because -1 is impossible. The exponential distribution is only defined for values of x >= 0. However, the probability that X = 2 (the mean of the distribution) is also equal to zero - but obviously possible.

Okay, but this just seems to be the "paradox" all over again. The dart can hit the point with a probability of 0.

Logically, if something is impossible, then it's probability is 0, is always true (as is the contrapositive). The converse - If the probability is zero, then it is impossible - and the inverse - if it is possible, then it's probability is not 0 - are not necessarily true.

aa

What I put in bold seems to imply that it's impossible for the dart to hit an intended point.

 

[ryan]

That's a fairly loose definition of zero. It's just calling a non-zero number zero because it's pretty close to zero.

..which we do all the time, when we're taking measurements of things. A 2 x 4 is not precisely 2" by 4" down to the angstrom. Even the device used to measure has uncertainty in it.

aa

I'm not seeing how that is relevant to the logic forum. The question clearly implies mathematically ideal objects.

In an ideal world we would stick to the point but we're not in one. Or maybe we are?

It's also relevant to point at the similarities of language and the traps that involves.

Also, the OP seems to implies a purely logical question only relevant to an ideal world yet it seems clear to me that ryan is mixing in a fair amount of realistic possibilities, not just in the OP but in subsequent posts, showing I think that he hopes to be able to deduce something about the material world from his paradox. The OP is about our representation(s) of reality I think.
EB

No, I am just studying math, and my mind runs wild.

Why can't it just be that if we live in the necessary ideal universe, then the probability of hitting a point with a dart is infinitesimal, and the chances of hitting a target that doesn't exist is 0?

 

[Speakpigeon]

So does an "impossible probability of zero" equate to an "almost impossible probability of zero"? Or, theoretically, is there an infinitesimally small probability?

There is only one infinitisemal small number: 0.

We are unable to specify any particular number which would be infinitesimally small yet not nul.

Metaphysically, you can conceive of them. Ah, the power of the metaphysical mind
EB

 

[Speakpigeon]

Why can't it just be that if we live in the necessary ideal universe, then the probability of hitting a point with a dart is infinitesimal, and the chances of hitting a target that doesn't exist is 0?

Sorry I can't answer properly a question that does make sense to me.

To me, in the ideal world you have described, the probability for any particular point to be hit by the dart is not zero. It is undetermined as 1/infinity is. You can go a bit further. The summation over the board should be 1 but we can't prove that it is because infinity/infinity is also undertermined. Yet, all this is a consequence of your specification of the ideal universe. It is unclear to me that such a universe could exist at all, meaning that if your specifications don't lead to a conclusive description of how things behave in it, it's not really a specification of any possible universe. IOW, the probability of your ideal universe to represent a real universe is zero.

So you say it is a paradox but to solve it you just need to prove that the summation of probabilities over the board gives 1 as expected. So it is a mathematical problem, not a logical paradox.

Intuitively, though, as long as 1/infinity is not zero, I don't see any problem with the expected outcome that one point would be picked up by the dart. However, since it is not a real universe, we cannot check whether any point would be picked up in reality. So, we only have our expectation that this would be the outcome. You want to see a paradox because you claim without proof that infinitesimally small is the same as zero. Sorry, prove it! Without this there is no paradox, just the mathematical problem of describing probabilities properly in this imaginary universe.
EB

 

[ryan]

Why can't it just be that if we live in the necessary ideal universe, then the probability of hitting a point with a dart is infinitesimal, and the chances of hitting a target that doesn't exist is 0?

Sorry I can't answer properly a question that does make sense to me.

To me, in the ideal world you have described, the probability for any particular point to be hit by the dart is not zero. It is undetermined as 1/infinity is. You can go a bit further. The summation over the board should be 1 but we can't prove that it is because infinity/infinity is also undertermined. Yet, all this is a consequence of your specification of the ideal universe. It is unclear to me that such a universe could exist at all, meaning that if your specifications don't lead to a conclusive description of how things behave in it, it's not really a specification of any possible universe. IOW, the probability of your ideal universe to represent a real universe is zero.

So you say it is a paradox but to solve it you just need to prove that the summation of probabilities over the board gives 1 as expected. So it is a mathematical problem, not a logical paradox.

Intuitively, though, as long as 1/infinity is not zero, I don't see any problem with the expected outcome that one point would be picked up by the dart. However, since it is not a real universe, we cannot check whether any point would be picked up in reality. So, we only have our expectation that this would be the outcome. You want to see a paradox because you claim without proof that infinitesimally small is the same as zero. Sorry, prove it! Without this there is no paradox, just the mathematical problem of describing probabilities properly in this imaginary universe.
EB

I call it a paradox because 0 = impossible, and 0 = unlikely. But impossible does not equal unlikely.

 

[Alcoholic Actuary]

That's a fairly loose definition of zero. It's just calling a non-zero number zero because it's pretty close to zero.

..which we do all the time, when we're taking measurements of things. A 2 x 4 is not precisely 2" by 4" down to the angstrom. Even the device used to measure has uncertainty in it.

aa

I'm not seeing how that is relevant to the logic forum. The question clearly implies mathematically ideal objects.

In an ideal world we would stick to the point but we're not in one. Or maybe we are?

It's also relevant to point at the similarities of language and the traps that involves.

Also, the OP seems to implies a purely logical question only relevant to an ideal world yet it seems clear to me that ryan is mixing in a fair amount of realistic possibilities, not just in the OP but in subsequent posts, showing I think that he hopes to be able to deduce something about the material world from his paradox. The OP is about our representation(s) of reality I think.
EB

Thank you, agreed. It's relevant because if we were just dealing with mathematical ideals, the discussion would have concluded by now. The entire hang-up seems to be when we apply the theory to the real world.

aa

 

[Alcoholic Actuary]

Why can't it just be that if we live in the necessary ideal universe, then the probability of hitting a point with a dart is infinitesimal, and the chances of hitting a target that doesn't exist is 0?

Sorry I can't answer properly a question that does make sense to me.

To me, in the ideal world you have described, the probability for any particular point to be hit by the dart is not zero. It is undetermined as 1/infinity is. You can go a bit further. The summation over the board should be 1 but we can't prove that it is because infinity/infinity is also undertermined. Yet, all this is a consequence of your specification of the ideal universe. It is unclear to me that such a universe could exist at all, meaning that if your specifications don't lead to a conclusive description of how things behave in it, it's not really a specification of any possible universe. IOW, the probability of your ideal universe to represent a real universe is zero.

So you say it is a paradox but to solve it you just need to prove that the summation of probabilities over the board gives 1 as expected. So it is a mathematical problem, not a logical paradox.

Intuitively, though, as long as 1/infinity is not zero, I don't see any problem with the expected outcome that one point would be picked up by the dart. However, since it is not a real universe, we cannot check whether any point would be picked up in reality. So, we only have our expectation that this would be the outcome. You want to see a paradox because you claim without proof that infinitesimally small is the same as zero. Sorry, prove it! Without this there is no paradox, just the mathematical problem of describing probabilities properly in this imaginary universe.
EB

I call it a paradox because 0 = impossible, and 0 = unlikely. But impossible does not equal unlikely.

The square root of 4 = 2, and the square root of 4 = -2. But 2 does not equal -2. Is that supposed to be some kind of paradox?

aa

 

[Alcoholic Actuary]

So does an "impossible probability of zero" equate to an "almost impossible probability of zero"? Or, theoretically, is there an infinitesimally small probability?

No. It is a special definition. Through the construction of the experiment, we first determine what is possible and what is impossible. All things impossible we assign a probability of zero - they occupy none of the probability distribution's sample space. If our theoretical example is best defined using an exponential distribution with a mean of 2, then the probability that X = -1 is equal to zero by definition, because -1 is impossible. The exponential distribution is only defined for values of x >= 0. However, the probability that X = 2 (the mean of the distribution) is also equal to zero - but obviously possible.

Okay, but this just seems to be the "paradox" all over again. The dart can hit the point with a probability of 0.

It can, because a probability of zero does not imply impossibility.

Logically, if something is impossible, then it's probability is 0, is always true (as is the contrapositive). The converse - If the probability is zero, then it is impossible - and the inverse - if it is possible, then it's probability is not 0 - are not necessarily true.

aa

What I put in bold seems to imply that it's impossible for the dart to hit an intended point.

Re-read it. I'm trying to say is that the part of the logic argument you've bolded is not always true. The probability that the dart hits an intended point is zero (theoretically), but not impossible.

Let's go back to the real number line. We have a continuous distribution defined over the real numbers and you are throwing darts at it. What is the probability you hit pi? It isn't 'infinitessimal'. It is zero.

aa

 

[ryan]

Why can't it just be that if we live in the necessary ideal universe, then the probability of hitting a point with a dart is infinitesimal, and the chances of hitting a target that doesn't exist is 0?

Sorry I can't answer properly a question that does make sense to me.

To me, in the ideal world you have described, the probability for any particular point to be hit by the dart is not zero. It is undetermined as 1/infinity is. You can go a bit further. The summation over the board should be 1 but we can't prove that it is because infinity/infinity is also undertermined. Yet, all this is a consequence of your specification of the ideal universe. It is unclear to me that such a universe could exist at all, meaning that if your specifications don't lead to a conclusive description of how things behave in it, it's not really a specification of any possible universe. IOW, the probability of your ideal universe to represent a real universe is zero.

So you say it is a paradox but to solve it you just need to prove that the summation of probabilities over the board gives 1 as expected. So it is a mathematical problem, not a logical paradox.

Intuitively, though, as long as 1/infinity is not zero, I don't see any problem with the expected outcome that one point would be picked up by the dart. However, since it is not a real universe, we cannot check whether any point would be picked up in reality. So, we only have our expectation that this would be the outcome. You want to see a paradox because you claim without proof that infinitesimally small is the same as zero. Sorry, prove it! Without this there is no paradox, just the mathematical problem of describing probabilities properly in this imaginary universe.
EB

I call it a paradox because 0 = impossible, and 0 = unlikely. But impossible does not equal unlikely.

The square root of 4 = 2, and the square root of 4 = -2. But 2 does not equal -2. Is that supposed to be some kind of paradox?

aa

Sneaky, but I think you're wrong. The square root of 4 is plus/minus 2. Can b = c, d = c but d doesn't equal b? I think this is the very first proof that I learnt.

 

[ryan]

That's a fairly loose definition of zero. It's just calling a non-zero number zero because it's pretty close to zero.

..which we do all the time, when we're taking measurements of things. A 2 x 4 is not precisely 2" by 4" down to the angstrom. Even the device used to measure has uncertainty in it.

aa

I'm not seeing how that is relevant to the logic forum. The question clearly implies mathematically ideal objects.

In an ideal world we would stick to the point but we're not in one. Or maybe we are?

It's also relevant to point at the similarities of language and the traps that involves.

Also, the OP seems to implies a purely logical question only relevant to an ideal world yet it seems clear to me that ryan is mixing in a fair amount of realistic possibilities, not just in the OP but in subsequent posts, showing I think that he hopes to be able to deduce something about the material world from his paradox. The OP is about our representation(s) of reality I think.
EB

Thank you, agreed. It's relevant because if we were just dealing with mathematical ideals, the discussion would have concluded by now. The entire hang-up seems to be when we apply the theory to the real world.

aa

So far Einstein is winning the debate on a continuous space-time over a discrete space-time. Replace "dartboard" with "location".

 

[Speakpigeon]

I call it a paradox because 0 = impossible, and 0 = unlikely. But impossible does not equal unlikely.

The square root of 4 = 2, and the square root of 4 = -2.
aa

Sneaky, but I think you're wrong. The square root of 4 is plus/minus 2.

Whoa! Zero is definitely to mark your grammar for both of you!!

If 2 and -2 are square roots of 4 then you should say a square root of 4 is 2 (not the square root) and a (or another) square root of 4 is -2.

Good grammar saves time and lives.

The question is whether infinitesimally small is equal to zero. Maybe it is in the real word, although even there I don't see why it wouldn't be something else like 10^-12 or something (see also "quantum"). In the universe postulated by ryan, infinitesimally small is not equal to zero. So, it depends what you want to argue about, the ideal world postulated by ryan or the real world?
EB