Can the probability of an event ever be exactly zero?

[krikker]

Probability is just a mathematical theory, and something we use to model things in 'the real world'. When using continuous pdfs via Lebesgue (can never recall the spelling) measure, the measure of a point of the real line is zero. There really is no way round that as it a simple 'fact'. If you want to argue that this isn't the correct model for whatever your situation is, krikker, then go ahead. It doesn't alter the fact that the (Lebesgue) measure of a point is 0.

I haven't really worked with that, but as I understand it, it is used to develop an algebra which includes infinites and infinitesimals. Therefore the measure can be defined as zero, so long as the algebraic operations are based on this definition. I'm not claiming this to be true, I'm in a bit of deep water. Just thought I remembered some of what you said from an old conversation with a mathematician friend of mine.

\frac 1 2 + \frac 1 4 + \cdots = \sum_{n=1}^{\infty}\frac 1{2^n}

I didn't know this relation to actually be true, since the right side is defined as the limit of the left side. If, however it holds true instead of being just a convention, then I can only express my joy since I use it myself all the time. Thought it was just one of those things engineers did and real mathematicians hated them for. Btw, it makes me even happier because my high school math teacher, who I disliked, made it a point how that relation didn't hold.

 

[matt grime]

What do you mean by 'actually true'? (In relation to the sum.) It is certainly true in the real numbers from the definition. Whether you consider that to be an 'actual truth' about 'reality' is almost certainly philosophical.

 

[krikker]

My point is that by definition
\sum_{n=1}^{\infty}
is
\lim_{N\to\infty} \sum_{n=1}^{N}

I thought that mathematically you couldn't put an equal sign between an endless sum and the limit of that sum. I knew people did it, but I thought it to be a convention rather than 100% correct.

 

[matt grime]

It is essentially the definition of the real numbers that they are equal (since one is *defined* to be the other). Of course it is non-trivial to show that this makes sense. Fortunately, we can construct the real numbers in many ways - dedeking cuts, equivalence classes of cauchy sequences, or equivalence classes of decimal expansions (or any other base than 10).

 

[D H]

One way to define the real numbers is via infinite series (they are called Cauchy sequences; google that phrase). It is non-mathematicians, not mathematicians, who have a problem with saying
\sum_{n=1}^{\infty} \frac 1 {2^n} \equiv \lim_{N\to\infty}\sum_{n=1}^N \frac 1 {2^n} = 1
Non-mathematicians (and high school math teachers, boo!) are the ones who say 0.999\cdots \ne 1. The limit of a sequence, if it exists, is a specific number. The limit does not differ from this number by some infinitesimal amount. It is the number.

What makes some mathematicians cringe is lack of rigor. One place where this occurs is the non-rigorous use of infinitesimals by engineers and physicists. Mathematicians switched from Newton's infinitesimals to the epsilon-delta notation formalism developed by Weierstrass because this formalism is mathematically sound. Physicists clung to the shorthand infinitesimals because they work (mostly). Mathematicians finally made the concept of infinitesimals rigorous in the 1960s by means of the non-standard analysis (another phrase to google). Anything that is true in standard real and complex analysis is true in the non-standard analyses. In particular
\left(\sum_{n=1}^{\infty}\frac 1 {2^n}\right)-1=0
whether one is using standard or non-standard analysis.

 

[krikker]

That's great. Thanks for the long explanation. Never thought it to be a contradiction, just some tiny area that math could never truly reach.

The one thing I still don't entirely understand is why you never see x=infinity if it is known to be exactly that.

 

[D H]

Infinity is not a number. Yet another thing every mathematician worth his or her salt knows.

 

[DeaconJohn]

Infinity is not a number. Yet another thing every mathematician worth his or her salt knows.

DH,

I am suprised at you. Whether or not infinity is a number is a matter of definition, from the viewpoint of the field of mathematics.

There are different definitions. It is a matter of preference, or perhaps of convenience.

----------------------------------------------------------------

When my oldest daughter Charlotte was six, she was scheduled to take an entrance exam to get into second grade in a private school.

I had stong suspicions that she would be asked what was the largest number she knew.

With mischief in my heart, I taught her to answer "infinity."

The time for the test came.

The tester reported to us that when she asked Charlotte what was the biggest number she knew, Charlotte smiled a big confident smile and answered,

"Infinity! And, infinity is equal to ten, because ten is the biggest number I know!"

:)

-------------------------------------------------------------------------

But, in all seriouness, how about this for a "universal" definition of a number ----

"A number is an answer to the question 'how many elements are in that set?' "

DJ

 

[DeaconJohn]

One way to define the real numbers is via infinite series (they are called Cauchy sequences; google that phrase). It is non-mathematicians, not mathematicians, who have a problem with saying
\sum_{n=1}^{\infty} \frac 1 {2^n} \equiv \lim_{N\to\infty}\sum_{n=1}^N \frac 1 {2^n} = 1
Non-mathematicians (and high school math teachers, boo!) are the ones who say 0.999\cdots \ne 1. The limit of a sequence, if it exists, is a specific number. The limit does not differ from this number by some infinitesimal amount. It is the number.

What makes some mathematicians cringe is lack of rigor. One place where this occurs is the non-rigorous use of infinitesimals by engineers and physicists. Mathematicians switched from Newton's infinitesimals to the epsilon-delta notation formalism developed by Weierstrass because this formalism is mathematically sound. Physicists clung to the shorthand infinitesimals because they work (mostly). Mathematicians finally made the concept of infinitesimals rigorous in the 1960s by means of the non-standard analysis (another phrase to google). Anything that is true in standard real and complex analysis is true in the non-standard analyses. In particular
\left(\sum_{n=1}^{\infty}\frac 1 {2^n}\right)-1=0
whether one is using standard or non-standard analysis.

DH, That is a beautiful explanation. I'm really impressed.

 

[DeaconJohn]

Zeno's paradox

While philosophers might be vexed by Zeno's paradox, mathematicians most definitely are not because any mathematician worth their salt knows that
\frac 1 2 + \frac 1 4 + \cdots = \sum_{n=1}^{\infty}\frac 1{2^n} = 1

Right on, DH!

Yeah, Zeno's paradox is difficult for modern mathematicians to understand. I will try to explain the paradox and try to explain while it is difficult for us moderns to understand at the same time. I will do this by giving a modern formulation of the paradox that shows where Zeno thought differently from the way we do.

The paradox is this.

Suppose a hare is chasing a rabbit and the hare goes twice as fast as the rabit.

The hare starts out 120 miles behind the rabbit and the hare runs at 60 miles an hour. The rabbit runs at 30 miles an hour. The race is track is four miles long.

After one hour, the hare is one half mile behind the rabbit. After another half hour the hare is 1/4 mile behind the rabbit.

The nth time period has length one hour / 2^n.

As each time period passes, the hare halves his distance to the rabbit.

To actually catch the rabbit, the hare would have to run for as long as it takes to transverse an infinite number of these intervals.

But, in the real world, infinity does not exist.

Hence the hare never catches the rabbit.

But we all know that the hare passes the rabbit when they have both run for one hour, and after two hours, the hare crosses the finish line.

This material is quasi-original with me, so, it may well stand improvement. Please let me know if it does.

There are too many dirrerent ways to explain this in modern terms to make a start on it. But, I think what I have written catches the essential idea and illustrates the different viewpoints of the relationship between mathematics and the real world.

Actually, the viewpoint of mathematicians even two centuries ago was closer to the Greeks.

DJ

"By quasi-original," I mean,

"I claim no credit for being the original proporter of these ideas in case they should happen to be right, but, I accept full responsibility for them if they are wrong."

 

[D H]

DH,

I am suprised at you. Whether or not infinity is a number is a matter of definition, from the viewpoint of the field of mathematics.

There are different definitions. It is a matter of preference, or perhaps of convenience.

You're right; I should have said infinity is not a real number. There are other definitions where infinity is a number -- the extended real number line, for example.

But, in all seriouness, how about this for a "universal" definition of a number ----

"A number is an answer to the question 'how many elements are in that set?' "

These are the cardinal numbers. What set has half of an element? pi elements? 1-i elements?

 

[HallsofIvy]

DH,

I am suprised at you. Whether or not infinity is a number is a matter of definition, from the viewpoint of the field of mathematics.

What D H meant to say is that infinity is not a member of the set of a real numbers (or complex numbers for that matter). That is what is normally understood by "numbers" with no other adjectives. As far as the real or complex numbers are concerned, whether or not infinity is a member is NOT a matter of preference or of convenience.

There are different definitions. It is a matter of preference, or perhaps of convenience.

----------------------------------------------------------------

When my oldest daughter Charlotte was six, she was scheduled to take an entrance exam to get into second grade in a private school.

I had stong suspicions that she would be asked what was the largest number she knew.

With mischief in my heart, I taught her to answer "infinity."

The time for the test came.

The tester reported to us that when she asked Charlotte what was the biggest number she knew, Charlotte smiled a big confident smile and answered,

"Infinity! And, infinity is equal to ten, because ten is the biggest number I know!"

:)

-------------------------------------------------------------------------

But, in all seriouness, how about this for a "universal" definition of a number ----

"A number is an answer to the question 'how many elements are in that set?' "

DJ

 

[DeaconJohn]

You're right; I should have said infinity is not a real number. There are other definitions where infinity is a number -- the extended real number line, for example.

Oh. My bad.

Please accept my apology, DH.

English is not a "context-free" language, and I mis-interperted the the context in which you were making your statement. In other words, it was "my bad," not "your lack," that resulted in my suprise.

These are the cardinal numbers. What set has half of an element? pi elements? 1-i elements?

Right-on, DH. My suggestion that this be a "universal" definition changed the "context" that required me to specify that I was only talking about "cardinal numbers."

My last paragraph should have read:

"But, in all seriouness, how about this for a "universal" definition of a (cardinal) number ----

"A cardinal number is an answer to the question, 'How many elements are in a set?' "

I really would like to know your opinion on this proposal. For example, this proposal would exclude the cardnalities of classes from the collection of cardinal numbers.

[A Note for Beginners: If I remember correctly, Bertrand Russel introduced "classes" into Zermelo-Frankel (ZF) set theory for the purpose of getting his paradox (Russell's paradox) out of set theory. If I understand correctly, one can consider the "class of all sets" in Russel's extension of ZF. In other words, a "class" is like a "really big set." Nobody cared - (correction - no mathematicians cared) - except those working in mathematical logic - until recent years when category theory began to take over.]

[Another Note for Beginners: The ubiquity of category theory in modern mathematics is a movement kicked off by Alexander Grothendieck's application of category theory to derive his generalization of the Hirzebuch-Riemann-Roch theorem in the 1950's. Category theory was developed (mostly by Grothendieck) in the thousands of pages of tomes called EGA and SGA that stand for the French equivalents of "Exposition of" - and "Seminar in" - "Algebraic Geometry." There were other giants - such as Grothendieck's advisor Jean Dieudonne, the mysterious Jean Paul Serre, and the incredible Pierre Deligne - whose mathematical work played major roles in bringing category theory to the attention of the mathematical community.]

Another example of the kind of thing I am asking about is whether or not the infinities in "non-standard analysis" match up with the infinite cardinal numbers in the usual set theory that mathematicians use. In other words, did Robinson include - or try to include - the cardinalities of classes in his "non-standard analysis."

(I have the impression that there are logical problems that are best avoided by excluding the sizes of classes from the collection of cardinal numbers. I don't remember what they are, except, i have the impression that they are related to Russel's paradox.)

(I'm guessing, here, that the collection of cardinal numbers form a class and not a set. I would like to know if that is correct.)

 

[DeaconJohn]

Right on, DH!

Yeah, Zeno's paradox is difficult for modern mathematicians to understand. I will try to explain the paradox and try to explain while it is difficult for us moderns to understand at the same time. I will do this by giving a modern formulation of the paradox that shows where Zeno thought differently from the way we do.

The paradox is this.

...

This material is quasi-original with me, so, it may well stand improvement. Please let me know if it does.

There are too many dirrerent ways to explain this in modern terms to make a start on it. But, I think what I have written catches the essential idea and illustrates the different viewpoints of the relationship between mathematics and the real world.

Actually, the viewpoint of mathematicians even two centuries ago was closer to the Greeks.

DJ

"By quasi-original," I mean,

"I claim no credit for being the original proporter of these ideas in case they should happen to be right, but, I accept full responsibility for them if they are wrong."

Sorry, guys, looks like I'm bringing up a subject that has already been wrung dry. And, been "wrung dry" by you guys in the not too distant past. Ah, one of the dangers of being a newbee.

However, I think that the viewpoint that I propose has a different spin than the viewpoints expressed in your previous posts. I am proposing that the crux of Zeno's reasoning (expressed in modern language) is that he didn't believe that an infinite quantities of things really existed in the real world.

Assuming that is what he was thinking, he might have been right of course, in which case one explanation of his paradox is that you can't keep dividing time in half. When you get down to a certain granularity, you take an instantaneous jump across the atomic length of time.

If I remember correctly, the Greeks did believe that matter was not infinitely divisible, that's what they (e.g., Lucretius?) meant by by "atom," the smallest, indivisible, unit of matter. Granted (a correction to my previous post) they did think that time had no beginning, so, my statement that they did not believe that infinity existed in the real world was a little hasty. But, it does not seem unreasonable to me that they thought that an infinity of intervals of time did not exist.

This seems more reasonable to me than the supposition that they thought that the sum of the geometric series diverged. More accurately, that they thought that it was impossible for an infinite number of terms to have a finite sum. Even though that is "literaly" what Zeno said, I suspect he took his "paradox" as a demonstration that infinite quantities do not exist i nature.

What do you all think? I know that at least some of yon haave thought about it.

Here is a pointer to one of your posts that suggests a different intrepertation than the one that I am purporting.

https://www.physicsforums.com/showthread.php?t=173095&highlight=Zeno's+paradox

Yours,

DJ

P.S.

It's important to consider what the ancients thought. They were excellent mathematicians (Archimedes is credited with discovering the essence of the calculus) and they had a different "worldview" than we do. So, from an objective viewpoint (like for example from the viewpoint of Bayesian probability theory), there is a non-zero probabiity that they wer more right than we are in their conception of realiity.

 

[D H]

This seems more reasonable to me than the supposition that they thought that the sum of the geometric series diverged. More accurately, that they thought that it was impossible for an infinite number of terms to have a finite sum. Even though that is "literaly" what Zeno said, I suspect he took his "paradox" as a demonstration that infinite quantities do not exist i nature.

What do you all think? I know that at least some of yon haave thought about it.

My first thought is that we are getting way off topic here.

My second thought is that Zeno simply thought any series with an infinite number of terms had to diverge, and yet it obviously doesn't. Paradox. Not surprising since he predates the concept of a convergent infinite series by a mere 2100 years or so.

 

[DeaconJohn]

My first thought is that we are getting way off topic here.

My second thought is that Zeno simply thought any series with an infinite number of terms had to diverge, and yet it obviously doesn't. Paradox. Not surprising since he predates the concept of a convergent infinite series by a mere 2100 years or so.

Thanks, DH!

I guess you're right about getting a little "off topic." Time to move on!

DJ

 

[Archosaur]

The next person to say "1 / infinity = 0" gets a complimentary punch in the face.

You might as well say "1 / applesauce = 0"

Because applesauce is just as much a number as infinity.

 

[quadraphonics]

The next person to say "1 / infinity = 0" gets a complimentary punch in the face.

http://en.wikipedia.org/wiki/Extended_real_number_line"

You might as well say "1 / applesauce = 0"

Because applesauce is just as much a number as infinity.

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

 

[BWV]

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?

 

[CRGreathouse]

0, naturally.

 

[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.