New Paradox: Zero Probability But Not Impossible?!

[Juma]

I'm pretty sure infinities are defined as uncountable, so I think your use of "countable infinite number" might be a bit off, but I could be wrong.

Natural numbers are countable infinite. As opposed to real numbers that are not countable.

 

[ryan]

There is a good analogy on Wikipedia using a dart board http://en.wikipedia.org/wiki/Almost_surely .

Imagine a perfectly smooth universe, dense like the natural numbers. There is a countable infinite number of points on a dart board. So the probability of hitting a particular point is 1/infinity which equals 0. Yet it is possible to hit the point with a dart that has a tip that comes to a point at the end of it. How can this be?

\( \frac{1}{\infty} = \epsilon \) with \(\epsilon\) being the symbol for an infinitesimal, which isn't 0.

But what does an infinitesimally small probability mean then? I thought that a probability had to be a real number.

I'm pretty sure infinities are defined as uncountable, so I think your use of "countable infinite number" might be a bit off, but I could be wrong.

Aleph-null is the only countable infinity. It's one-to-one with the naturals.

And lets say that you're going to hit the dartboard no matter what. Well, you're going to hit some point with 100% probability. And it's going to be one of those infinite number of points.

It's not 100% if the thrower is aiming for a specific point. Let's assume that the dart thrower is really bad, and he has an equal opportunity at hitting any point on the board. His technique is so bad that the dart lands in a totally random place on the board every time.

 

[Kharakov]

There is a good analogy on Wikipedia using a dart board http://en.wikipedia.org/wiki/Almost_surely .

Imagine a perfectly smooth universe, dense like the natural numbers. There is a countable infinite number of points on a dart board. So the probability of hitting a particular point is 1/infinity which equals 0. Yet it is possible to hit the point with a dart that has a tip that comes to a point at the end of it. How can this be?

\( \frac{1}{\infty} = \epsilon \) with \(\epsilon\) being the symbol for an infinitesimal, which isn't 0.

But what does an infinitesimally small probability mean then? I thought that a probability had to be a real number.

Well, the sum of all the probabilities has to equal 1. So there is still an infinitesimal chance that the dart will hit the correct spot. But it's basically zero, assuming there are no other influencing factors, like someone influencing events for you.

I'm pretty sure infinities are defined as uncountable, so I think your use of "countable infinite number" might be a bit off, but I could be wrong.

Aleph-null is the only countable infinity. It's one-to-one with the naturals.

Thanks.

 

[Speakpigeon]

There is a good analogy on Wikipedia using a dart board http://en.wikipedia.org/wiki/Almost_surely .

Imagine a perfectly smooth universe, dense like the natural numbers. There is a countable infinite number of points on a dart board. So the probability of hitting a particular point is 1/infinity which equals 0. Yet it is possible to hit the point with a dart that has a tip that comes to a point at the end of it. How can this be?

First, it's not how the material universe seems to work, mainly because we don't know point-like things, or anything being one- or two-dimensional. Everything is three-dimensional and has a finite volume. In the case of dart games, the smallests elements making up the board has a finite probability of being hit by the dart, which is itself not point-like. Further, "hits" in the real world involve complex interactions that the situation you discribe doesn't take into account so that even if dart and board particles were point-like, the probability of the dart hitting one on the board would still not be zero. So, it's a paradox but only a theoretical paradox.

Second, I think that there is no solution to your problem in mathematical terms. The reason has to be that you cannot specify what would be a random trajectory of the dart and you cannot specify points on the board without somehow specifying particular points. Therefore you cannot determine the probability of its landing. Not even the probability that the dart actually hits a point of the board. Whether the dart lands on the board and then on which particular point and with what probabilities would depend on how you specify the universe where the dart and the board exist.

It's like Santa Claus.
EB

 

[rizdek]

This seems to be a variety of the many Zeno's paradoxes. Maybe it could be called Zeno's Zero Probability Paradox?

Once it is thrown, it no longer has the chance of landing anywhere, it is destined for a specific point. But per the hypothetical setup, even that specific point isn't real, it still contains an infinite number of "points." Eventually when it gets to within say, a Planck length of the surface of the square (whether it represents the universe or not seems superfluous) one has to decide if there can still be infinite points within an area smaller than a Planck length. If so, in my mind it becomes similar to the dilemma, "you can't get from point A to point B because there are infinite number of points between the points one has to traverse to get from A to B. But it does open a new idea to me about the various Zeno paradoxes in that if there are infinite number of points between two points, then we are looking at a universe with infinite points and how can we even think of point A as having a location if even within the tiniest "area" there are still, theoretically, an infinite number of points?

I've been told by folks well versed in calculus that Zeno's paradoxes are answered with calculus...so maybe a person with a strong background in math can explain this paradox too.

OTOH, it is obvious that when you throw a dart in this universe, it does land somewhere, we frequently get from point A to point B and faster folks catch up to and pass slower folks. So the problem must be in the conceptualizing of the problem rather than on reality itself.

 

[beero1000]

 Countable additivity and  Measure theory. Why should area/volume/measure be additive over an uncountable index set?

I used the countable infinity to make it easier.

Instead of making it easier, you made it wrong. You have not described a well-defined probability measure.

I was hoping that the general point would be made.

A human aims at a point and always hits either the point or an area of an infinite number of points surrounding the intended target. And if the dart is not going to hit the point, then let's just assume that the dart has an equal opportunity of hitting anywhere within, let's say, a centimeter of the intended target.

That is where countable additivity comes in. It is perfectly consistent to have each point hit with probability 0 while at the same time the probability that some point is hit is 1. If you don't want to deal with probabilities, how about areas?

Hmmm, I am not sure what your issue is here.

What is the area of a point? What is the area of a square? Can you get the area of the square by adding up the areas of all the points?

I don't think that I can answer the first or the last question with the level of calculus that I am at.

I know. That is why I asked the questions.

You are trying to say that for a collection of disjoint sets \(A_i\), it is true that \(P(\bigcup_{i \in I} A_i) = \sum_{i\in I} P(A_i)\). This is only true if the index set I is countable. This notion is called  countable additivity, and is part of the definition of a probability measure (for good reason!).

 

[Speakpigeon]

But ryan dart board is a countable set of points so the right member of your equation is defined if we assume that each Ai is a point. In this case, the summation of an infinite number of probabilities each equal to 1/infinite is, what, undefined? Or just 1? If the latest, then your equation would be verified. I would guess it's the former since as you are adding one after the other it stays at zero so the limit should be zero. Which would be my intuitive reply for the question of the sum of the areas of all the points of a 1m2 square.

Yet, isn't that the point of the paradox, that it doesn't add up?
EB

 

[beero1000]

But ryan dart board is a countable set of points so the left member of your equation is defined if we assume that each Ai is a point. In this case, the summation of an infinite number a probabilities each equal to 1/infinite is, what, undefined? Or just 1? If the latest, then your equation would be verified. I would guess it's the former since as you are adding one after the other it stays at zero so the limit should be zero. Which would be my intuitive reply for the question of the sum of the areas of all the points of a 1m\(2\) square.

He has not defined a valid probability measure on the dartboard, which is why his problem statement is wrong. He tried to simplify the idea by moving to a countable dartboard, but just ended up making it self-contradictory. He takes the points \(A_i\) on the dartboard and says \(P(A_i) = 0\) for all i. A probability measure requires that \(P(\bigcup_{i\in I} A_i) = 1\). By countable additivity, both cannot simultaneously be true.

If you go back to the original, uncountable board, you can easily have \(P(A_i) = 0\) for all i, and \(P(\bigcup_{i\in I} A_i) = 1\). This is only surprising if you expected \(P(\bigcup_{i \in I} A_i) = \sum_{i \in I} A_i\) no matter what index set you use. That is why I brought up the area example (IMO area is more intuitive than probability) - why should we expect area to work that way?

Yet, isn't that the point of the paradox, that it doesn't add up?
EB

There's no reason to expect it to add up. A point has zero area and a unit square has area one. Can we say that the area of the square is the sum of the areas of the points?

 

[Speakpigeon]

Yet, isn't that the point of the paradox, that it doesn't add up?

There's no reason to expect it to add up. A point has zero area and a unit square has area one. Can we say that the area of the square is the sum of the areas of the points?

I don't think so, but why precisely?


Also there is no good reason to rely on an analogy between the case of areas and the case of probabilities. Here, I guess it comes down to the same thing since the probability that the dart picks a particular point is a function of the area of the point but that's just a particular case.

I take your point that we need a reason to expect addibility. But assuming that the dart can only pick a point of the board no matter what, the probability of that is 1. But the probability of any particular point to be picked is 1/infinity and the right member of your equation sums it up over all points of the board. We expect this summation to give an overall probability of 1 just because it's that of the dart picking a point of the board. If not, then some other outcome would have to have a probability above zero, maybe that the dart goes through without picking a point. But excluding that, we still don't seem to know how to add up 1/infinity over an infinity of points so we cannot prove the expected result.

Yet, we still expect that it's the correct one. Isn't that a reason to expect addibility? Isn't it the intuitive result given the set-up described? In other words, the only barrier to solving the paradox may be that we don't know how to add up 1/infinity in this case.
EB

 

[none]

ryan, I am wondering if you think that the dart board is an infinite number of atoms, wouldn't that make a difference if there were a finite number of atoms?

 

[beero1000]

Yet, isn't that the point of the paradox, that it doesn't add up?

There's no reason to expect it to add up. A point has zero area and a unit square has area one. Can we say that the area of the square is the sum of the areas of the points?

I don't think so, but why precisely?

Uncountable additivity would imply that x = y for any two numbers x and y, (in the square example, it argues that 0 = 1). That is the 'paradox', and why it is not interesting/useful/allowed.

Also there is no good reason to rely on an analogy between the case of areas and the case of probabilities. Here, I guess it comes down to the same thing since the probability that the dart picks a particular point is a function of the area of the point but that's just a particular case.

Area and probability are essentially the same - they are both measures. People have much better intuition with respect to areas, which is why I tried to shift away from probability.

I take your point that we need a reason to expect addibility. But assuming that the dart can only pick a point of the board no matter what, the probability of that is 1. But the probability of any particular point to be picked is 1/infinity and the right member of your equation sums it up over all points of the board. We expect this summation to give an overall probability of 1 just because it's that of the dart picking a point of the board. If not, then some other outcome would have to have a probability above zero, maybe that the dart goes through without picking a point. But excluding that, we still don't seem to know how to add up 1/infinity over an infinity of points so we cannot prove the expected result.

Yet, we still expect that it's the correct one. Isn't that a reason to expect addibility? Isn't it the intuitive result given the set-up described? In other words, the only barrier to solving the paradox may be that we don't know how to add up 1/infinity in this case.
EB

1/infinity is nonsensical.

 

[steve_bnk]

Consider a continuous probabilitydistribution such as a Gaussian or ''bell curve'.

Between the two end points of the distribution there are an infinite number of points. The probability of any point on the curve is zero. The probability of one event outof an infinite number of possibilities is zero.

The probability at a point is determined by integrating, taking the area, under the curve around the point. The integral between two identical numbers is zero.

In the real world there are no infinite physical conditions. A real surface is comprised f a finite numberof elements.

Given a board of an infinite number ofpoints the probability of a finitely bounded area could be calculated. In the limit as dx*dy gets small the area approaches theprobability of a point


Ryan periodically tries to make psychical reality assertions based on the mathematical abstraction o fa continuous number line with an infinite number of points.

 

[Loren Pechtel]

A dart isn't infinitely small.

In reality the probability of hitting any given point (assuming a random hit) is the area of the dart point : the area of the board.

 

[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

 

[Tom Sawyer]

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.

 

[ryan]

ryan, I am wondering if you think that the dart board is an infinite number of atoms, wouldn't that make a difference if there were a finite number of atoms?

I don't understand your question.

 

[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

 

[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?

 

[beero1000]

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

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

 

[Tom Sawyer]

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.