# Probabilities with infinities

1. Dec 20, 2011

### sri sharan

A probability of zero for an event says that it is impossible right?Then lets say that I need to choose a number randomly among the infinite numbers between 2 and 5.The probability of picking a particular number should be zero.And yet i can pick a number randomly(whose probobility of being picked should be zero).This looks like a contradiction of sorts.Is something wrong with my definition of zero probability??

2. Dec 20, 2011

### Bacle2

No, you are correct; the probability of choosing any one number , or, equivalently, the measure of any one point is 0. For disjoint events ai

But measure is additive, tho not countably additive, i.e., you can conclude that
Ʃm(ai)=m(a1)U(a2)U.......U(an)

This is not true for uncountable sums (the only way an uncountable sum converges is if there are only countably-many non-zero terms). I think there are more subleties here; maybe someone else can comment.

3. Dec 20, 2011

### sri sharan

I didn't doubt that before. what I actually meant was that my definition of zero probability as an impossible event might be wrong.I feel a more subtle and precise definition might exist which will account for this here.My apologies if I have not been clear before.

4. Dec 20, 2011

### Bacle2

In my understanding, only possible outcomes are considered in the sample space, and these events in the sample space and no others are assigned a probability, whether zero or not. So, I think a probability of zero does not denote impossibility. With a frequentist definition of probability, probability zero would denote that , as the number of throws increases, the relative frequency goes to zero, with some variant of convergence.

5. Dec 20, 2011

### Stephen Tashi

That's a permissible interpretation for a probability distribution that has discrete values. Probability densities that are defined on a continuum of values, can have a non-zero value at a particular number, but if you do the integration of a typical probability density function f(x) from x = a to x = a you naturally get zero. The fact that a point has zero length is not an obstacle to assigning length to an interval. The fact that a point has "zero probability" is not an obstacle to assigning a finite probability to an interval. There is no mathematical contradiction in that regard.

If X is a random variable that is uniformly distributed on [2,5] then probability texts permit themselves to pose problems where you are to imagine a specific sample drawn from that distribution, for example, X = 2.356. The fact that they ask you to imagine such an example is not proof that this is a possible event. Whether such an event is really possible or not is a question about reality and, if any mathematics bears on the question, it is not the mathematics taught in ordinary probability courses.

To imagine that we take a sample from the uniform distribution on [2,5] is imagine something that is impossible in practical terms. If we are taking a physical measurement, we only measure precisely to a finite number of decimal places. So our measurement only defines a possible interval for the measured value. If we simulate samples using a computer, we likewise only get finite precision. So when you say "I can pick a number randomly" it is not clear that this is possible since you are speaking of picking a number with infinite precision.

Last edited: Dec 20, 2011
6. Dec 20, 2011

### sri sharan

Ok all this is totally confusing and not clarifying anything. Can anyone give a meaning of what zero probability means when continuous variables and probability densities are involved

7. Dec 20, 2011

### sri sharan

But that is just our practical inability of picking a complete random number. That can't inhibit the mathematics.

8. Dec 20, 2011

### SW VandeCarr

What is the probability of choosing a real number on the (continuous) interval [0,1]? It has to be zero since the real numbers are dense in the interval. However, obtaining such a number is possible. This simply means that any non zero probability we assign to such a number does not define an interval that contains only that number. Zero is the limit value of the probability that can be assigned to selecting a unique number on the real interval [0,1]. Perhaps this article will help.

http://www.statlect.com/subon/probab2.htm

9. Dec 20, 2011

### sri sharan

Ok that cleared up things a bit, but I still cant find an exact quantitative definition of what zero probability means,not in that article nor any where else on the net.Can some one provide a definition of the same

10. Dec 20, 2011

### sri sharan

11. Dec 20, 2011

### Stephen Tashi

The practical impossibility of exact sampling doesn't impede mathematics, it only impedes your attempt to find a paradox! You (and the blog you linked) are stating a quandary that mixes statements from mathematics with statements about the real world. (I assume you are using the word "possible" as it is used in common speech, to say something about reality.)

The mathematical definition of an event with zero probability is simply that a probability measure assigns it probability zero. There is no commentary on whether such events are possible in the real world or not. Standard probability theory does not prove or disprove that it is possible to take exact samples from a uniform distribution on [2,5] (which are events with zero probability) It merely states consequences that apply (such as the Central Limit Theorem) if we assume such samples are taken. Mathematics states the consequences of certain assumptions and definitions. It does not deal with whether the assumptions themselves are true or false in any objective sense or whether the definitions correspond to anything in reality.

By analogy, plane geometry does not comment on the existence of lines and points in the real world. The fact that a measure of length assigns length zero to a point is not a commentary on the existence or non-existence of points in the real world.

If you want to tackle the real world, you must deal with physics, not pure math.

One way to transform your question into a purely mathematical context, is to drop the assumption that we can take exact samples from a uniform distribution on [2,5]. Replace this with the assumption that there is a an infinite sequence of measurement tecniques, each with finite precison, but such that the sequence becomes arbitrarily precise. (Think of it as imagining all future technologies.) Take some probability theorem, such as the Central Limit Theorem. Replace the hypothesis of that theorem with the hypothesis that we can only sample with some finite precision. Examine what the consequences of this are as we increase the precision using the more refined measurement techniques. What is the limiting behavior of the result? Does it match the conclusion of the theorem when exact sampling is assumed?

I've never seen this worked out in a textbook. Perhaps it was one of those "exercise left to the reader" things that I neglected to do.

12. Dec 20, 2011

### lavinia

probability zero does not mean that it is impossible to choose a number. It means that on average you will never choose it.

13. Dec 20, 2011

### Bacle2

Sri Sharan wrote, in part:

" Ok that cleared up things a bit, but I still cant find an exact quantitative definition of what zero probability means,not in that article nor any where else on the net.Can some one provide a definition of the same "

Well, I'm sorry I cannot give you a definition without a choice of probability perspective: if you are a frequentist (I think this is the term) , then an event has probability 0 is one so that, as the number of trials approaches ∞ , the relative frequency of the event , i.e., the ratio of successes to total trials, approaches zero. Maybe someone knows what this may mean in a Bayesian model.

Is this what you are looking for?

The mathematical aspects of measure zero are:
i)∫xxdx=0 (this is equivalent to saying individual points have
measure zero)

ii) The sum of uncountably-many non-zero terms (here probabilities of individual points in a continuum) will not converge , let alone sum to 1 (note that an integral over a continuous interval is a sort of countably-infinite sample from an uncountable set).

14. Dec 21, 2011

### lavinia

Technically a set of probability zero has measure zero. Probability zero and measure zero mean the same thing.

But the strong law of large numbers says that the average value of the characteristic function of a subset converges to the measure of the subset. This means that if you sample your entire probability space with a sequence of independent selections and give yourself a 1 if you select from the subset and a zero otherwise then the average value of this sequence converges almost surely to the measure of the set. For a set of measure zero it will converge to zero.

Last edited: Dec 21, 2011
15. Dec 22, 2011

### sri sharan

Hmm, ok this makes sense.Now everything is clear