New Paradox: Zero Probability But Not Impossible?!

[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?

 

[none]

because the surface area of the dart board is finite.

 

[ryan]

because the surface area of the dart board is finite.

But there is an infinite number of possible places that the dart could have landed on, and it will land on one of them. So if the dart thrower tries to hit a specific point, the dart may actually hit that point. It has to hit some point; why not the one chosen?

 

[none]

because the surface area of the dart board is finite.

But there is an infinite number of possible places that the dart could have landed on, and it will land on one of them. So if the dart thrower tries to hit a specific point, the dart may actually hit that point. It has to hit some point; why not the one chosen?

you probably have to argue the dart is infinitely small, and how do you know the dart board is a dart board when it is just a collection of particles?
LOL

 

[beero1000]

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

 

[ryan]

because the surface area of the dart board is finite.

But there is an infinite number of possible places that the dart could have landed on, and it will land on one of them. So if the dart thrower tries to hit a specific point, the dart may actually hit that point. It has to hit some point; why not the one chosen?

you probably have to argue the dart is infinitely small, ...

Not necessarily, imagine that we zoom in on the tip of the dart to the extent that it looks round and no longer sharp. Even a round circle that hits a perfectly flat surface must make contact with only a point first. And by "point", I mean a true point with zero dimensions. This is actually provable by using calculus.

 

[ryan]

 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.

 

[none]

because the surface area of the dart board is finite.

But there is an infinite number of possible places that the dart could have landed on, and it will land on one of them. So if the dart thrower tries to hit a specific point, the dart may actually hit that point. It has to hit some point; why not the one chosen?

you probably have to argue the dart is infinitely small, ...

Not necessarily, imagine that we zoom in on the tip of the dart to the extent that it looks round and no longer sharp. Even a round circle that hits a perfectly flat surface must make contact with only a point first. And by "point", I mean a true point with zero dimensions. This is actually provable by using calculus.

how do you know the tip of the dart is a perfect circle in shape?
because you made it that way right?
it's all theoretical right?

 

[ryan]

because the surface area of the dart board is finite.

But there is an infinite number of possible places that the dart could have landed on, and it will land on one of them. So if the dart thrower tries to hit a specific point, the dart may actually hit that point. It has to hit some point; why not the one chosen?

you probably have to argue the dart is infinitely small, ...

Not necessarily, imagine that we zoom in on the tip of the dart to the extent that it looks round and no longer sharp. Even a round circle that hits a perfectly flat surface must make contact with only a point first. And by "point", I mean a true point with zero dimensions. This is actually provable by using calculus.

how do you know the tip of the dart is a perfect circle in shape?
because you made it that way right?
it's all theoretical right?

Yeah, we are assuming a dense and ideal universe.

 

[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.

 

[Tom Sawyer]

If I'm throwing the dart, it doesn't matter what the probabilities are. I'm still always missing the point I'm aiming at.

 

[ryan]

 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.

 

[ryan]

If I'm throwing the dart, it doesn't matter what the probabilities are. I'm still always missing the point I'm aiming at.

I am not sure if you are joking. If you are serious, then I would say that it must hit some point. So why can't it be the targeted point? Any other point is just as likely.

 

[none]

...
Yeah, we are assuming a dense and ideal universe.

sorry but I really didn't mean to sound flippant.
what about Planck length in your calculations? a smallest measurement... or smallest distance possible, right?

 

[Tom Sawyer]

If I'm throwing the dart, it doesn't matter what the probabilities are. I'm still always missing the point I'm aiming at.

I am not sure if you are joking. If you are serious, then I would say that it must hit some point. So why can't it be the targeted point? Any other point is just as likely.

Clearly, you've never seen me throw a dart. I was once playing with my sister (in a game of darts - get your mind out of the gutter, you perv ) and managed to hit her in the back with the dart. And no - she was not somehow between me and the dart board and I was actually aiming at the dart board.

Do whatever fancy maths you like, my lack of aim overrides any equations you can come up with.

 

[fast]

I find the very notion of infinity to be semi-applicable in nature, meaning while things like numbers can easily be understood as infinite, the application of such a concept to certain arena's of the physical world would test our ability to remain reasonable in the application of it ... But that's just an idea I'm mulling over.

 

[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? 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?

 

[ryan]

 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.

 

[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.
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.

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.

 

[Juma]

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?

What is the problem? This post states no contradiction. (But a vague and possbly misleading use of "possible ")