New Paradox: Zero Probability But Not Impossible?!

[Speakpigeon]

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

But even in a continuous space-time the probability for a point on the board to be hit depends on the size of the tip of the dart. Suppose we assume as you did that the probability of the dart hitting the board is 1 and now suppose that the tip of the dart has the size of the board... Then each point will be hit.
The real world may not be like you think...
EB

 

[ryan]

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

Why would you want to discuss something outside of what my OP specifically states? If you want this question posed in a real world scenario, then just start a new thread.

 

[ryan]

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

But even in a continuous space-time the probability for a point on the board to be hit depends on the size of the tip of the dart. Suppose we assume as you did that the probability of the dart hitting the board is 1 and now suppose that the tip of the dart has the size of the board... Then each point will be hit.
The real world may not be like you think...
EB

I put that in the OP as well, "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". Why should I even bother with details if people don't read them.

 

[beero1000]

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

But even in a continuous space-time the probability for a point on the board to be hit depends on the size of the tip of the dart. Suppose we assume as you did that the probability of the dart hitting the board is 1 and now suppose that the tip of the dart has the size of the board... Then each point will be hit.
The real world may not be like you think...
EB

I put that in the OP as well, "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". Why should I even bother with details if people don't read them.

+1

And to answer your question, the standard (axiomatic, measure theoretic) approach to probability makes no distinction between never and almost never. Both correspond to probability 0, and that's all there is to it.

However, mathematicians are an industrious lot and so, of course, there are extensions of these notions of probability that make the distinction between them. In particular, you can talk about infinitesimal probabilities, or about an infinite number of iterations, etc, and these will give some sense of the difference between never and almost never.

For example, take your dartboard and throw 2^c darts at it (where c is the cardinality of the continuum). Then, each point expects an infinite number of darts to hit them, so they clearly can't be 'never' situations.

 

[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

Why would you want to discuss something outside of what my OP specifically states? If you want this question posed in a real world scenario, then just start a new thread.

I was addressing myself to both you and Alcoholic Actuary. I was only pointing out that you are not talking about the same thing. You say you are talking about an ideal world you specified mathematically while he is talking about our actual material world or some scientific model of it.
EB

 

[Speakpigeon]

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

But even in a continuous space-time the probability for a point on the board to be hit depends on the size of the tip of the dart. Suppose we assume as you did that the probability of the dart hitting the board is 1 and now suppose that the tip of the dart has the size of the board... Then each point will be hit.
The real world may not be like you think...
EB

I put that in the OP as well, "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". Why should I even bother with details if people don't read them.

I read the OP and you didn't read my post properly.

You were arguing from Einstein so you could only be talking about the actual material world or his mathematical model of it, not the ideal world you specified in the OP. In your ideal world, dart tips may be point-like but in the actual material world they are not, not as far as we know since interactions require energy and energy exchanges are quantified.

You should hire a continuity manager...
EB

 

[Alcoholic Actuary]

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

But even in a continuous space-time the probability for a point on the board to be hit depends on the size of the tip of the dart. Suppose we assume as you did that the probability of the dart hitting the board is 1 and now suppose that the tip of the dart has the size of the board... Then each point will be hit.
The real world may not be like you think...
EB

I put that in the OP as well, "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". Why should I even bother with details if people don't read them.

+1

And to answer your question, the standard (axiomatic, measure theoretic) approach to probability makes no distinction between never and almost never. Both correspond to probability 0, and that's all there is to it.

Agreed.

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

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.

The point is that arriving at the same answer from 2 different directions doesn't create a paradox. I can arrive at a probability of zero by definition - any result outside of the probability distribution will be assigned a probability of zero. These results are not possible. I can also arrive at a probability of zero by calculation - any result inside of the probability distribution that has 0 area under it's density function.

You've done the same thing in the dart board. The probability my dart hits any point inside the dartboard is zero - by calculation. It isn't impossible, it justs contains zero mass of the probability distribution. The probability my dart hits outside of the dartboard is also zero - by definition. No part of the distribution exists outside of the dartboard, therefore it is impossible to throw a dart outside of the dartboard.

Outside of the dartboard does not equal inside of the dartboard, yet both results are zero. It's not a paradox.

aa