# Infinity and probability **URL:** \ **Category:** Great Debates\ **Created:** [November 6, 2003, 11:11am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684 "2003-11-06T11:11:36Z")\ **Posts on this page:** 20\ **Page:** 4
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 12, 2003, 3:19pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/61 "2003-11-12T15:19:54Z")
> [@](#): > > \*Originally posted by Cabbage \* > \*\*Of course, it may be possible that doing this would force you to give up certain other “nice” properties of a standard probability function, but it’s not clear to me that that would necessarily be the case. \*\* As soon as you’re dealing with hyperreals you lose completeness, which may or may not be important. It’d be interesting to see this done in detail. ---
**Author:** ![Sergio](https://avatars.discourse-cdn.com/v4/letter/s/b77776/32.png) [@Sergio](https://boards.straightdope.com/u/Sergio)\ **Post date:** [November 12, 2003, 3:41pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/62 "2003-11-12T15:41:07Z")
> [@](#): > > \*Originally posted by Cabbage \* > \*\*With the standard definition of a probability function, there ain’t nothing you can do about it. > > However (and I’m no probability theorist), I imagine you could extend the notion of a probability function P. Instead of having P map into the real interval [0,1], have P map into the hyperreal interval [0,1] (or some other extension of the reals that includes infinitesimals). The idea, of course, being that sets which previously got mapped to zero, yet were still “possible”, would get mapped to mapped to an infinitesimal, while sets that were previously mapped to one, yet not “certain”, would get mapped to a number infinitesimally less than one. > > In fact, I would be damn surprised if this _hasn’t_ been researched before. Of course, it may be possible that doing this would force you to give up certain other “nice” properties of a standard probability function, but it’s not clear to me that that would necessarily be the case. \*\* I don’t see where this would lead us! As I pointed in a previous post, in a continuous pdf, the probability of a particular outcome is zero, but the probability of the outcome liyng in a determined interval is generally nonzero. Since the width of the interval can be made arbitrarily small, I don’t see what you need infinitesimals for. ---
**Author:** ![Shade](https://avatars.discourse-cdn.com/v4/letter/s/2bfe46/32.png) [@Shade](https://boards.straightdope.com/u/Shade)\ **Post date:** [November 12, 2003, 3:46pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/63 "2003-11-12T15:46:27Z")
**Cabbage** : You’re right, I was too hasty. Let’s step back from this argument a moment. Can I say unarguably that (1) Depending on what form the infinite universe takes it can be ‘almost’ certain or ‘almost’ impossible that there’s another copy of the OP somewhere and (2) one way of illustrating this is with the standard mathematical definition of probability? Maybe that isn’t actually the best way to undersand it, come to think of it… ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 12, 2003, 3:48pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/64 "2003-11-12T15:48:14Z")
> [@](#): > > \*Originally posted by Sérgio \* > \*\*I don’t see where this would lead us! As I pointed in a previous post, in a continuous pdf, the probability of a particular outcome is zero, but the probability of the outcome liyng in a determined interval is generally nonzero. Since the width of the interval can be made arbitrarily small, I don’t see what you need infinitesimals for. \*\* As **Cabbage** mentioned, it would allow you to assign different probabilities to events that must happen and events that have probability 1 now but aren’t necessary. It’s more of an interesting exercise than anything. ---
**Author:** ![Sergio](https://avatars.discourse-cdn.com/v4/letter/s/b77776/32.png) [@Sergio](https://boards.straightdope.com/u/Sergio)\ **Post date:** [November 12, 2003, 4:04pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/65 "2003-11-12T16:04:10Z")
> [@](#): > > \*Originally posted by Shade \* > \*\*Is there any other way to do it? > > Take a simple example. Suppose there’s a completely unkown physical constant between 0 and 1. There are some things we can deduce about the universe if it is in certain ranges. Is there a way to assign probabilities to these? > > You can say ‘No can do. Sorry.’ but if you actually want to answer questions like the OP, I think it’s necessary to assume some notion of probability, and I can’t think of a useful one that would work in this case without having 0-prob ‘possible’ outcomes. > > What would you have us do? \*\* If you know what is the pdf of the values of this constant, you can without any problem to assign a probability to the event that its value lies in the interval (a;b), 0\<=a\<=b\<=1. For instance, if all values in the interval [0;1] are equiprobable, you have a uniform pdf and the probability that the value of your constant belongs to the interval is simply b-a. With other pdf’s the calculation is less straightforward, but still possible. You may have to use integral calculus, but you don’t need to pe a PhD to do this. If you have no idea of what the pdf is, there is no means, to my knowledge, to assign any probability to the event. ---
**Author:** ![Shade](https://avatars.discourse-cdn.com/v4/letter/s/2bfe46/32.png) [@Shade](https://boards.straightdope.com/u/Shade)\ **Post date:** [November 12, 2003, 4:28pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/66 "2003-11-12T16:28:50Z")
**Sérgio** : But, can you assign a probability to both the open and closed intervals? If not, which is ok? If so, mustn’t they be the same and doesn’t that fail the ‘intuitive’ test Ryan was talking about? ---
**Author:** ![Sergio](https://avatars.discourse-cdn.com/v4/letter/s/b77776/32.png) [@Sergio](https://boards.straightdope.com/u/Sergio)\ **Post date:** [November 12, 2003, 6:25pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/67 "2003-11-12T18:25:29Z")
**Shade** The probability for the closed interval is equal to the probability for the open interval plus the probabilities of the limits of the interval: p{[a;b]} = p{(a;b)} + p{a} + p{b} = p{(a;b)} + 0 + 0 = p{(a;b)} So it does not matter if the interval is open or closed, the probability is the same! ---
**Author:** ![Shade](https://avatars.discourse-cdn.com/v4/letter/s/2bfe46/32.png) [@Shade](https://boards.straightdope.com/u/Shade)\ **Post date:** [November 12, 2003, 6:30pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/68 "2003-11-12T18:30:56Z")
Sorry, perhaps I got a bit ahead of myself. I assumed that having two nested events with the same probabilty was one of the things your system was trying to avoid. Further, we have the problem that there are events U=[a,b] and V=(a,b) and we can find the probability either happening, but you can’t say “the probability of V not U.” That ommission is looks very inelegant to me. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 12, 2003, 6:40pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/69 "2003-11-12T18:40:20Z")
Since V is a subset of U, P(U and not V) = P(U) - P(V). It’s 0. ---
**Author:** ![Shade](https://avatars.discourse-cdn.com/v4/letter/s/2bfe46/32.png) [@Shade](https://boards.straightdope.com/u/Shade)\ **Post date:** [November 12, 2003, 6:51pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/70 "2003-11-12T18:51:06Z")
Yes, but sergio’s system was in response to my observation that ended > [@](#): > > I can’t think of a useful one that would work in this case without having 0-prob ‘possible’ outcomes. ---
**Author:** ![Sergio](https://avatars.discourse-cdn.com/v4/letter/s/b77776/32.png) [@Sergio](https://boards.straightdope.com/u/Sergio)\ **Post date:** [November 12, 2003, 6:59pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/71 "2003-11-12T18:59:27Z")
**Shade** p(V,~U) = p(a) + p(b) = 0 + 0 = 0. As I said before, the probability of a particular outcome in a continuous pdf is zero. Note that this refers to a priori probabilities. The a priori probability of a is zero, so the a priori probability of ~a is 1. If I perform the experiment and find a, the a posteriori probability of a is 1 and the a posteriori probability of ~a is 0. So, events with zero a priori probabilities do occur. Once they occur their a posteriori probabilities become one. I don’t know if it explains your doubt. ---
**Author:** ![Shade](https://avatars.discourse-cdn.com/v4/letter/s/2bfe46/32.png) [@Shade](https://boards.straightdope.com/u/Shade)\ **Post date:** [November 12, 2003, 7:15pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/72 "2003-11-12T19:15:57Z")
As far as I can see your probability system is exactly the same as the standard. And specifically includes possible events with zero probabilities. Your post was a response to my asking if it was possible to avoid this (as apparently Ryan desired.) _Was your post intended as an example of a definition of probability that didn’t involve 0-probability posssible events??_ If not, then I apologise, I misunderstood. Though imho understandably, since that’s what you replied to. Is so, I don’t see how it doesn’t! ---
**Author:** ![Sergio](https://avatars.discourse-cdn.com/v4/letter/s/b77776/32.png) [@Sergio](https://boards.straightdope.com/u/Sergio)\ **Post date:** [November 12, 2003, 7:49pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/73 "2003-11-12T19:49:32Z")
**Shade** I used the standard probability system. My point is: there is no need to postulate different systems, as the one proposed by Cabbage with infinitesimals. Events with zero probability are totally meaningful. If you need a very small nonzero probability in a continuous pdf all you need is to take a tiny interval.:smack: ---
**Author:** ![Shade](https://avatars.discourse-cdn.com/v4/letter/s/2bfe46/32.png) [@Shade](https://boards.straightdope.com/u/Shade)\ **Post date:** [November 12, 2003, 7:56pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/74 "2003-11-12T19:56:46Z")
OK, I apologise, I totally misunderstood. In my defense, you didn’t mention infitessimals anywhere, and defined the probability of an interval to be its length, and gave the probability of a point to be zero, so I don’t think it was quite clear that that’s what you meant. ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [November 12, 2003, 8:29pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/75 "2003-11-12T20:29:55Z")
> [@](#): > > My point is: there is no need to postulate different systems, as the one proposed by Cabbage with infinitesimals. Well, I guess that depends on how you define “need”. I do think that it’s somewhat “intuitively unsatisfying” that a probability function can’t distinguish between a set with probability one that _isn’t_ certain, from a set with probability one that _is_ certain. And regardless, I definitely think it would be interesting to see what results you could get by extending the notion of probability to include infinitesimals. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 12, 2003, 8:38pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/76 "2003-11-12T20:38:27Z")
> [@](#): > > \*Originally posted by Cabbage \* > \*\*Well, I guess that depends on how you define “need”. I do think that it’s somewhat “intuitively unsatisfying” that a probability function can’t distinguish between a set with probability one that _isn’t_ certain, from a set with probability one that _is_ certain. And regardless, I definitely think it would be interesting to see what results you could get by extending the notion of probability to include infinitesimals. \*\* Now that I think about it, I don’t think that “certain” is defined in a sufficiently rigorous manner to do what you wanted. I suppose you could arbitrarily say that some events are certain (and that certainty is preserved by unions and intersections), and that the complement of a certain event is impossible… Might be worth looking at. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 12, 2003, 8:59pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/77 "2003-11-12T20:59:08Z")
Now that I think about it, it’s not possible to do what **The Ryan** wants using infinitesimals. Consider the case of an infinite sequence of coin flips–the limit of 1/2[sup]n[/sup] as n grows unboundedly is 0, and the introduction of infinitesimals won’t change that. ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [November 12, 2003, 9:12pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/78 "2003-11-12T21:12:37Z")
> [@](#): > > Consider the case of an infinite sequence of coin flips–the limit of 1/2[sup]n[/sup] as n grows unboundedly is 0, and the introduction of infinitesimals won’t change that. Wouldn’t it be correct to say that the limit of 1/2[sup]n[/sup] is an infinitesimal? ---
**Author:** ![Sergio](https://avatars.discourse-cdn.com/v4/letter/s/b77776/32.png) [@Sergio](https://boards.straightdope.com/u/Sergio)\ **Post date:** [November 12, 2003, 9:36pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/79 "2003-11-12T21:36:44Z")
> [@](#): > > \*Originally posted by Cabbage \* > \*\*Wouldn’t it be correct to say that the limit of 1/2[sup]n[/sup] is an infinitesimal? \*\* No, the limit is zero! ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 12, 2003, 9:45pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/80 "2003-11-12T21:45:10Z")
> [@](#): > > \*Originally posted by Cabbage \* > \*\*Wouldn’t it be correct to say that the limit of 1/2[sup]n[/sup] is an infinitesimal? \*\* I seriously doubt it–I can construct a sequence of hyperreals with the nth term corresponding to 1/2[sup]n[/sup] (the nth term is (1/2[sup]n[/sup], 1/2[sup]n[/sup], 1/2[sup]n[/sup], …)), and the limit as n goes to infinity of that is (0, 0, 0, …), a hyperreal that corresponds to 0. It’s certainly possible that you know more about infinitesimals than I do and can see a flaw in the above reasoning; if so, please explain it. [Previous page](https://boards.straightdope.com/t/infinity-and-probability/211684.md?page=3) [Next page](https://boards.straightdope.com/t/infinity-and-probability/211684.md?page=5)