Can the probability of an event ever be exactly zero?

[HallsofIvy]

The point being that 0 probability on a continuous probability distribution does NOT mean "impossible" nor does probability 1 mean "certain to happen".

 

[Archosaur]

True, if by "applesauce" you mean "some element of the extended/projective/hyper-real numbers that has greater magnitude than any finite real number."

A greater magnitude? Infinity doesn't have a magnitude! I do not mean your proposed jargon!

I mean applesauce!

 

[Archosaur]

*cough cough*

And don't cough at me.

If 1/infinity = 0 then 1 = 0 * infinity.
If 1/infinity = 0 then 2*1/infinity = 2*0, also known as 2/infinity = 0
If 2/infinity = 0 then 2 = 0 * infinity

If 1 = 0*infinity AND 2 = 0*infinity
Then 1=2

Reductio ad absurdum.

 

[quadraphonics]

A greater magnitude? Infinity doesn't have a magnitude!

The extended (and projective) real lines both contain elements with a larger magnitude than any finite number, called infinity.

You might also have heard of http://en.wikipedia.org/wiki/Transfinite_number" .

I do not mean your proposed jargon!

It's not my proposal. The extended and projective reals have been around since long before I was born.

I mean applesauce!

Unlike infinity, I have never encountered a number system that includes an element "applesauce."

And don't cough at me.

Did you read the link?

If 1/infinity = 0 then 1 = 0 * infinity.
If 1/infinity = 0 then 2*1/infinity = 2*0, also known as 2/infinity = 0
If 2/infinity = 0 then 2 = 0 * infinity

No, the expression 0*\infty is undefined in both the extended and projective real numbers, although 1/\infty=0 does indeed hold in both of them.

 

[CRGreathouse]

The extended (and projective) real lines both contain elements with a larger magnitude than any finite number, called infinity.

Is that true? In the extended reals, certainly, but in the projectives?

 

[quadraphonics]

Is that true? In the extended reals, certainly, but in the projectives?

I believe so, although the proviso of larger magnitude is important. It is certainly not the case that projective infinity is larger than (or smaller than) any real number, but it should still be the case that projective infinity is farther from 0 than any real number.

I think the rub is that any meaningful way of defining the magnitude of projective infinity would have to give results in terms of extended infinity, since you must distinguish a negative magnitude (which is impossible) from a positive one. So I suppose my statement just reduces to a statement about the extended reals.

 

[JoeBonasia]

I think the simple answer to the original question was answered in one of the first replies...I didn't read every page so I apologize if I sound like an earlier response I just felt liek throwing my 2 cents out quick!

The assumption that you have an infinite number of events in the time interval is false...

Assuming it takes a finite interval for the event of pressing the button to occur, and assuming the total interval of opportunity is 2 minutes, then you will have a finite # of hypothetical events therefore the probaility should be:

1/(maximum cases of which the button prssing event can occur over the 2 minute interval) = finite number

This misleading situation is analagous to Zeno's paradox of space and time, I suggest you check out the book, I believe it is entitled Zeno's Paradox, it is a very interesting read.

 

[JoeBonasia]

So here is a similar problem, given an continuous uniform distribution over the interval [0,1], what is the probability that any number picked at random is rational?

now this is good!

 

[Archosaur]

No, the expression 0*\infty is undefined in both the extended and projective real numbers, although 1/\infty=0 does indeed hold in both of them.

But that's basic algebra!

 

[quadraphonics]

But that's basic algebra!

Indeed, the cost of including infinity in a number system is that certain basic expressions must remain undefined, when they include infinity. But so what? They're still defined as usual for every finite number, and the expressions where infinity is sensible (like 1/\infty=0) are still well defined.

There is also a finite number that does not work in many basic algebraic manipulations, but is nevertheless included in many standard number systems.

 

[Archosaur]

Indeed, the cost of including infinity in a number system is that certain basic expressions must remain undefined, when they include infinity. But so what? They're still defined as usual for every finite number, and the expressions where infinity is sensible (like 1/\infty=0) are still well defined.

There is also a finite number that does not work in many basic algebraic manipulations, but is nevertheless included in many standard number systems.

Can you give me an example of a finite number that I can't multiply both sides of an equation by?

I'm not trying to be a brat. I'm just not seeing this.

I would agree that the limit as n -> infinity of 1/n = zero

But I'm having a hard time with this 1/infinity = 0 thing...

 

[quadraphonics]

Can you give me an example of a finite number that I can't multiply both sides of an equation by?

No, but there is a finite real number that you can't divide both sides of an equation by :]

I would agree that the limit as n -> infinity of 1/n = zero

But I'm having a hard time with this 1/infinity = 0 thing...

\infty, as it is used in the extended reals, is essentially a shorthard for that type of limit expression.

 

[Archosaur]

\infty, as it is used in the extended reals, is essentially a shorthard for that type of limit expression.

I am totally okay with this! I'll sleep easy tonight, thanks!