# 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:** 2
**Author:** ![Bippy\_the\_Beardless](https://avatars.discourse-cdn.com/v4/letter/b/ac8455/32.png) [@Bippy\_the\_Beardless](https://boards.straightdope.com/u/Bippy_the_Beardless)\ **Post date:** [November 7, 2003, 2:00am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/21 "2003-11-07T02:00:08Z")
**Shade** is it ever possible to actually pick a random number (or any number) between 0 and 1. The number would have to be defined to infinite precision. Can we say that “Chosing digits randomly to make an infinite string” is even theoretically possible. Are we really justified in saying that any given string can be infinitely long, and can be compared to another infinitely long string of digits? Thinking again of an infinite Universe, is the state of that Universe in its entirity, one of an infinite set of zero probability states it may have been, or is it a single state of probability = 1 ? ---
**Author:** ![Shade](https://avatars.discourse-cdn.com/v4/letter/s/2bfe46/32.png) [@Shade](https://boards.straightdope.com/u/Shade)\ **Post date:** [November 7, 2003, 2:37am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/22 "2003-11-07T02:37:35Z")
> [@](#): > > But if, for instance, you pick a random number between 0 and 1, then, as the probability of picking any specific number is zero, but some number is picked. That was my parsing of the statement that “The Lebesgue measure of any point is zero, but of the unit interval is one,” which I think is how we were taught to interpet it. I don’t know of any system for actually providing a random number. But I think that mathematicians define probability in terms of measures , so from their point of view this should be as valid as picking a number from 1,2,3,4,5,6. But I’m not sure I completely understand your post; am I getting warmer? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 7, 2003, 4:12am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/23 "2003-11-07T04:12:11Z")
**Bippy** is talking about physically picking a real number between in [0, 1]. Sure, that’s impossible because of the limits of finite precision. But we can still talk about it, and that’s what we’re doing. > [@](#): > > \*Originally posted by Pedro \* > \*\*DAMN this board is impossible today. Sorry about the double post. To make my arguments mathematicaly: > > Definition of the event _a_: getting tails on n consecutives tries. > > P(a)= 1/2^n > > lim(n-\>infinite) P(a) = 0. > > So with infinite tries you will not have always tails, with absolute certainty. \*\* Only if probability 0 is equivalent to never happening. As has been mentioned before, that’s a bad assumption. Unfortunately, explaining this in any detail will get you into some rather advanced math very quickly. I can’t think of a good introduction that doesn’t assume the equivalent of a bachelor’s in math. \*\* > [@](#): > > I don’t make any claims here. But I could make the same argument with a random sequence of natural numbers. What exactly is this research trying to prove? That pi is random? Are you saying it is not enough that it has been proven that there isn’t any repeating pattern in pi? How about : > > a) If any finite sequence of natural numbers does not repeat (proven), there are infinitely many different sequences. > b) If there are infinititely many finite sequences, there is any finite sequence. > \*\* a) is true. b) is false–see the counterexample provided by **Jabba** The conclusion of b) is what people are trying to prove about pi. ---
**Author:** ![Czarcasm](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/czarcasm/32/4050_2.png) [@Czarcasm](https://boards.straightdope.com/u/Czarcasm)\ **Post date:** [November 7, 2003, 4:21am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/24 "2003-11-07T04:21:07Z")
If you flip a coin an infinite amount of times, then there is a small probability that it will come up heads every time, and a extremely large probability that tails will eventually come up, right? However, given an _infinite_ amount of time, all probabilities will eventually come up. Dores this mean that there is _no_ probability of a coin coming up heads every time in an infinite number of flips, or does it mean that the two contradictory possibilities are both possible? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 7, 2003, 4:31am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/25 "2003-11-07T04:31:51Z")
> [@](#): > > \*Originally posted by Czarcasm \* > **However, given an _infinite_ amount of time, all probabilities will eventually come up.** This sentence is false. ---
**Author:** ![Czarcasm](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/czarcasm/32/4050_2.png) [@Czarcasm](https://boards.straightdope.com/u/Czarcasm)\ **Post date:** [November 7, 2003, 4:48am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/26 "2003-11-07T04:48:25Z")
How is it possible to have a possible event _not_ eventually happen given an infinite amount of time? ---
**Author:** ![Czarcasm](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/czarcasm/32/4050_2.png) [@Czarcasm](https://boards.straightdope.com/u/Czarcasm)\ **Post date:** [November 7, 2003, 4:52am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/27 "2003-11-07T04:52:00Z")
Duh! By having one event cancel out the possibility of the other event. I is sooo smart sometimes. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 7, 2003, 4:55am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/28 "2003-11-07T04:55:16Z")
Even if there’s no cancellation, there’s no guarantee that an event will ever occur, no matter its possibility. Strange, but true. ---
**Author:** ![Czarcasm](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/czarcasm/32/4050_2.png) [@Czarcasm](https://boards.straightdope.com/u/Czarcasm)\ **Post date:** [November 7, 2003, 5:00am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/29 "2003-11-07T05:00:57Z")
Given an infinite amount of time and no cancellation, can you name an event of small probability that will never occur? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 7, 2003, 5:05am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/30 "2003-11-07T05:05:31Z")
No, I can’t. If an event is possible, then it may occur. However, that doesn’t guarantee that it _will_ occur. ---
**Author:** ![The\_Ryan](https://avatars.discourse-cdn.com/v4/letter/t/7feea3/32.png) [@The\_Ryan](https://boards.straightdope.com/u/The_Ryan)\ **Post date:** [November 7, 2003, 7:14am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/31 "2003-11-07T07:14:24Z")
ultrafilter > [@](#): > > But we can still talk about it, and that’s what we’re doing. Well, yeah, but “talking about it” and “rigourously defining it” are two different things. We can talk about the properties of the largest prime, but that doesn’t mean it exists. I think that the problem here is that we trying to assign a real number to the probability, but it’s not really a real number. For one thing, this space isn’t Hausdorff: given an infinite number of die rolls, getting at least one 6 and getting any sequence at all are both assigned the probability 1, but they are really two different numbers, even though they lack disjoint neighborhoods. ---
**Author:** ![JasonFin](https://avatars.discourse-cdn.com/v4/letter/j/4af34b/32.png) [@JasonFin](https://boards.straightdope.com/u/JasonFin)\ **Post date:** [November 7, 2003, 7:32am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/32 "2003-11-07T07:32:13Z")
Consider the following pseudo-program and an associated proposition: 1. Start off by picking any number P in the interval 0\P, go back to step 2. 4. Otherwise, stop. Proposition: The above program, if run, will eventually terminate. This program is analogous to looking for a particular event in an infinite universe. If the event has a nonzero probability P of occurring at any given place, we start by checking somewhere and seeing if it’s happening right there. If not, we check somewhere else (let’s say we use a space-filling pattern such that every location will eventually be checked). The proposition is equivalent to the statement that this event is guaranteed to eventually happen somewhere. I suspect that using only the traditional axioms of probability, this proposition is undecidable in the formal sense—it cannot be proven to be either true or false. To extend the system to answer questions of this nature, one would need to add an additional axiom equivalent to either this proposition or its negation. I can imagine that someone might feel, intuitively, either that this proposition is obviously true or that it is obviously false. I personally lean towards the idea that this proposition is neither true nor false; but then again, I don’t believe the Principle of the Excluded Middle ought to apply to the real world, and I suspect mainstream logicians would disagree. ---
**Author:** ![SentientMeat](https://avatars.discourse-cdn.com/v4/letter/s/43a26b/32.png) [@SentientMeat](https://boards.straightdope.com/u/SentientMeat)\ **Post date:** [November 7, 2003, 9:24am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/33 "2003-11-07T09:24:30Z")
Gosh, this thread went sideways quickly! First up, it is important not to think of this 4 thingy as a _number_, but as a _process_. Hence when we ask “what is the probability P of an event given an infinite sample size N?” we answer not by saying P=1 if N=4 but that P _tends to_ (6) 1 as N64. 4 is one of those stark reminders that, as Einstein put it, > [@](#): > > Insofar as the expressions of mathematics refer to reality they are not certain, and insofar as they are certain they do not refer to reality. ---
**Author:** ![Scarf-Ace](https://avatars.discourse-cdn.com/v4/letter/s/bc79bd/32.png) [@Scarf-Ace](https://boards.straightdope.com/u/Scarf-Ace)\ **Post date:** [November 7, 2003, 11:50am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/34 "2003-11-07T11:50:29Z")
Whoo too much maths for me already… I understand a little about probability (enough to read this thread without going ga-ga); I think if I was to formulate the OP in a more formal way i’d say “given an infinite universe, an event with an extremely low probability is no more likely to happen than in a finite universe.”. I think this is because in an infinite universe for any event E with a probability ~0 there is always an event G with a slightly higher probability. No? Ultrafilter: > [@](#): > > ``` > Originally posted by Czarcasm > However, given an infinite amount of time, all probabilities will eventually come up. > > ``` > > This sentence is false. Yah thats what i’m trying to say. ---
**Author:** ![The\_Great\_Unwashed](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_great_unwashed/32/4791_2.png) [@The\_Great\_Unwashed](https://boards.straightdope.com/u/The_Great_Unwashed)\ **Post date:** [November 7, 2003, 12:39pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/35 "2003-11-07T12:39:58Z")
Can I toss a spanner into these well-oiled works? What is this _probability_ of which you speak? Surely everything either _is_, or, _isn’t_? ---
**Author:** ![spingears](https://avatars.discourse-cdn.com/v4/letter/s/ebca7d/32.png) [@spingears](https://boards.straightdope.com/u/spingears)\ **Post date:** [November 7, 2003, 2:21pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/36 "2003-11-07T14:21:10Z")
. . . Ya ahl bin reedin 2 mux uh tht thr Hi Si Fi… ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 7, 2003, 4:01pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/37 "2003-11-07T16:01:07Z")
> [@](#): > > \*Originally posted by The Ryan \* > **Well, yeah, but “talking about it” and “rigourously defining it” are two different things. We can talk about the properties of the largest prime, but that doesn’t mean it exists.** We can also rigorously define the properties of the largest prime. What’s your point? \*\* > [@](#): > > I think that the problem here is that we trying to assign a real number to the probability, but it’s not really a real number. For one thing, this space isn’t Hausdorff: given an infinite number of die rolls, getting at least one 6 and getting any sequence at all are both assigned the probability 1, but they are really two different numbers, even though they lack disjoint neighborhoods. \*\* wtf? We have a function from the powerset of infinite sequences to [0, 1] that satisfies the Kolmogorov axioms, and that’s all that anyone requires for a probability distribution. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 7, 2003, 4:06pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/38 "2003-11-07T16:06:48Z")
> [@](#): > > \*Originally posted by The Ryan \* > **Well, yeah, but “talking about it” and “rigourously defining it” are two different things. We can talk about the properties of the largest prime, but that doesn’t mean it exists.** We can also rigorously define the properties of the largest prime. What’s your point? \*\* > [@](#): > > I think that the problem here is that we trying to assign a real number to the probability, but it’s not really a real number. For one thing, this space isn’t Hausdorff: given an infinite number of die rolls, getting at least one 6 and getting any sequence at all are both assigned the probability 1, but they are really two different numbers, even though they lack disjoint neighborhoods. \*\* wtf? We have a function from the powerset of infinite sequences to [0, 1] that satisfies the Kolmogorov axioms, and that’s all that anyone requires for a probability distribution. ---
**Author:** ![Sergio](https://avatars.discourse-cdn.com/v4/letter/s/b77776/32.png) [@Sergio](https://boards.straightdope.com/u/Sergio)\ **Post date:** [November 7, 2003, 4:16pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/39 "2003-11-07T16:16:50Z")
[QUOTE] \*Originally posted by ultrafilter \* \*\*We can also rigorously define the properties of the largest prime. What’s your point? \*\* No, we can not! There is no largest prime. This can be proved very easily. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 7, 2003, 4:33pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/40 "2003-11-07T16:33:00Z")
> [@](#): > > \*Originally posted by Sérgio \* > \*\* > > > [@](#): > > > > \*Originally posted by ultrafilter \* > > **We can also rigorously define the properties of the largest prime. What’s your point?** > > No, we can not! There is no largest prime. This can be proved very easily. \*\* p is the largest prime number iff p is a prime number, and for any prime number q, q \< p. If it weren’t rigorously defined, how could we prove it doesn’t exist? [Previous page](https://boards.straightdope.com/t/infinity-and-probability/211684.md?page=1) [Next page](https://boards.straightdope.com/t/infinity-and-probability/211684.md?page=3)