# 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:** 18\ **Page:** 7
**Author:** ![Sergio](https://avatars.discourse-cdn.com/v4/letter/s/b77776/32.png) [@Sergio](https://boards.straightdope.com/u/Sergio)\ **Post date:** [November 18, 2003, 3:47pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/121 "2003-11-18T15:47:10Z")
**Meta-Gumble** I am attacking the problem from a mathematical and not philosophical frame of mind. Let us momentarily forget what I said about experiences and focus only in genetics. Since there is a finite number of combinations of genes, given an infinite number of persons, there must be repetitions. We can state that at least one combination of genes will be shared by an infinite number of people. But we can say nothing about your specifical combination. May be you are unique. All that statistics can say is that at least one combination is shared by an infinite number of people with probability 1. It is impossible to establish the probability that your gene combination is shared by more than 1 person. We can only say that it is between 0 and 1. When we put the past experiences in the formation of a person we can say with certainty that the probability of another Meta-Gumble is zero. There is an infinite number of experiences anyone of us can have. Not necessarilly hard ones as falling in love or losing a cherished one. Soft experiences as hearing a song or watching a bird are part of your intelectual luggage and make you unique. ---
**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 18, 2003, 11:12pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/122 "2003-11-18T23:12:10Z")
But is it true that “There is an infinite number of experiences anyone of us can have” ? If we are only made up of quantized parts each with only finite number of possible states, then there is only a finite number of experiences anyone can have. The number is stupendously big, but it may well be finite, even if there exists some completely free states that can have any value (non-quantized) it still may be that there exist only a finite number of distinguishable experiences that anyone can have. So even though experiences may have differed in the smallest of minutae, they don’t differ in the effect they have on the individuals brain. ---
**Author:** ![Pedro](https://avatars.discourse-cdn.com/v4/letter/p/919ad9/32.png) [@Pedro](https://boards.straightdope.com/u/Pedro)\ **Post date:** [November 19, 2003, 1:21am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/123 "2003-11-19T01:21:27Z")
> [@](#): > > _Originally posted by **ultrafilter** _: > The fact that possible events have probability 0 is perfectly consistent with the frequentist interpretation: over any finite amount of time, we expect that event not to happen. Ah, but saying we expect the event not to happen is not the same as saying the event will not happen for certain, which is the conclusion I would expect to draw from P(a)=0. Which you are going to say is wrong and I accept that. > [@](#): > > I don’t know of any text offhand that addresses this, but as you may have noticed, all of the mathematicians here have accepted this point. > > The proper domain for dealing with infinite sample spaces is measure theory. I conclude that until I learn measure theory I’m not qualified to debate this so I will refrain from commenting further. We’d just be going around in circles. I didn’t mean to question your expertise, you obviously know a lot more about this than I do and I value your comments but it still does not carry the same weight as a formal argument. I’m just an engineering student with a strong interest in maths searching for enlightenment. Thank you for your input, and everyone else, too, even the mathematicians although they are obviously wrong. 😉 ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [November 19, 2003, 1:48am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/124 "2003-11-19T01:48:21Z")
**Pedro** , I mentioned (kind of in passing, so it may have been missed) earlier that my idea of extending the notion of a probability function to include infinitesimals in the range _has_ been researched. So if you’re uncomfortable with the unintuitive notion that “probability zero” is not synonymous with “impossible”, probability can be extended in such a way that our model assigns probability zero only to events that are impossible, while assigning an infinitesimal probability to events that previously had probability zero, yet were still possible (and, of course, the corresponding analogues for “probability one” vs. “certain”). This is known as _nonstandard probability_, which may be useful as a key word to search on if you’re interested in learning more. ---
**Author:** ![Sergio](https://avatars.discourse-cdn.com/v4/letter/s/b77776/32.png) [@Sergio](https://boards.straightdope.com/u/Sergio)\ **Post date:** [November 19, 2003, 9:00am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/125 "2003-11-19T09:00:44Z")
> [@](#): > > \*Originally posted by Bippy the Beardless \* > \*\*But is it true that “There is an infinite number of experiences anyone of us can have” ? > If we are only made up of quantized parts each with only finite number of possible states, then there is only a finite number of experiences anyone can have. The number is stupendously big, but it may well be finite, even if there exists some completely free states that can have any value (non-quantized) it still may be that there exist only a finite number of distinguishable experiences that anyone can have. So even though experiences may have differed in the smallest of minutae, they don’t differ in the effect they have on the individuals brain. \*\* Any one of us can have a finite number of experiences, but the universe of experiences is, IMO, infinite. Human beings are time variant systems. The same experience has different effects with time. For instance, seeing a huge spider can be a traumatic experience for a child, but is at most a disgusting one for a non-phobic adult. In a not so extreme example, the same experience will have different effect according to the state of mind of the experiencer. Supposing time non-quantized (wich is true outside the subatomic level), i think the universe of experiences is infinite. ---
**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 19, 2003, 4:24pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/126 "2003-11-19T16:24:59Z")
Even thoug time is not quantized, the human mind can only tell differences in time to about 1/50th second so whether an even occured a few milliseconds differently does not effect the person differently. So though the universe of experiences may be infinite the number of distinguishable experiences may non the less be finite. Looked at another way, the brain has a finite number of neurons, each neuron made up of a large number of atoms, the number of states that a brain can be in is limited because of its finite size, and finite number of distinguishable states. Here I use distinguishable to express that two events are distinguishable if they leed to different states at any time after the two events. So though a neuron firing at time t r time t+1 millisecond is distinguishable, if the complete chemical and excitation state of the brain for all time after either of the events is the same then both events are indistinguishable to the brain/neuron system and can I believe be considered as identical events. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 19, 2003, 4:31pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/127 "2003-11-19T16:31:18Z")
> [@](#): > > \*Originally posted by Pedro \* > \*\*I conclude that until I learn measure theory I’m not qualified to debate this so I will refrain from commenting further. We’d just be going around in circles. I didn’t mean to question your expertise, you obviously know a lot more about this than I do and I value your comments but it still does not carry the same weight as a formal argument. I’m just an engineering student with a strong interest in maths searching for enlightenment. Thank you for your input, and everyone else, too, even the mathematicians although they are obviously wrong. 😉 \*\* Actually, the couple paragraphs I gave on measure are the sketch of a formal argument–filling in the details would be tedious and not terribly enlightening, so I’ll leave them out. I’ve started to think that the problem here is trying to relate an uncountable sample space to physical reality. That just doesn’t work. ---
**Author:** ![Sergio](https://avatars.discourse-cdn.com/v4/letter/s/b77776/32.png) [@Sergio](https://boards.straightdope.com/u/Sergio)\ **Post date:** [November 19, 2003, 5:38pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/128 "2003-11-19T17:38:28Z")
**Bippy** I will accept your opinion about the finite number of possible experiences. I don’t know enough brain physiology to argue that. So, it is possible that there exists another **Meta-Gumble** in an infinite Universe. Probability theory does not warrant that a possible event must occur in an infinite number of events. If there is only a finite number of characteristics that define a given human being, in an infinite Universe there must exist an infinite number of repetitions. It is certain that a particular set of characteristics will be shared by an infinite number of persons, but nothing can be said about the set of characteristics that define **Meta-Gumble**. He may well be unique. And an arbitrary set of characteristics that you choose can very well be atribbuted to the empty set. No human being has these characteristics, even in an infinite Universe. ---
**Author:** ![Pedro](https://avatars.discourse-cdn.com/v4/letter/p/919ad9/32.png) [@Pedro](https://boards.straightdope.com/u/Pedro)\ **Post date:** [November 20, 2003, 3:42am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/129 "2003-11-20T03:42:01Z")
> [@](#): > > \*Originally posted by Cabbage \* > \*\ ***Pedro** , I mentioned (kind of in passing, so it may have been missed) earlier that my idea of extending the notion of a probability function to include infinitesimals in the range _has_ been researched. So if you’re uncomfortable with the unintuitive notion that “probability zero” is not synonymous with “impossible”, probability can be extended in such a way that our model assigns probability zero only to events that are impossible, while assigning an infinitesimal probability to events that previously had probability zero, yet were still possible (and, of course, the corresponding analogues for “probability one” vs. “certain”). This is known as _nonstandard probability_, which may be useful as a key word to search on if you’re interested in learning more. \*\* Yes, I didn’t miss it **Cabbage** , thanks. It seems very interesting, and beautiful solution too. I definitely plan to read on it after I finish [this](http://www.amazon.com/gp/reader/0198531710/ref=sib_dp_pt/104-8027747-9101518#reader-link) book and do some catching up on my nonstandard analysis. ---
**Author:** ![don\_t\_ask](https://avatars.discourse-cdn.com/v4/letter/d/e68b1a/32.png) [@don\_t\_ask](https://boards.straightdope.com/u/don_t_ask)\ **Post date:** [November 20, 2003, 4:55am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/130 "2003-11-20T04:55:52Z")
A few years ago I heard a scientist talking on the radio about the idea of infinite possibilities. He was essentially making the point that the “idea” of infinity led people to accept premises that were so unlikely as to be pointless. He used the monkeys, typewriters, complete works of Shakespeare model as his proof. He said that by definition enough monkeys with enough typerwriters would eventually type the complete works of Shakespeare (Monty Burns not withstanding) but what does this really mean? Say their typewriters have only letters and a space bar. We will capitalise and punctuate. The chances of randomly typing “To be or not to be tha” is 27 ^22 or 30,903,154,382,632,612,361,920,641,803,529 to 1. Now there are about 31,557,600 seconds in a year. We need about 979,261,869,807,355,830,669,019 monkey years to type this fragment of a speech. The generally accepted age of the Solar System is 4.55 billion years. So if our monkeys have been there for all time we need 215,222,388,968,649 monkeys. So his final point was, how useful is it to say that theoretically 215 trillion monkeys given all the time of recorded history could type a fragment of one speech from one play by Shakespeare. He seemed to believe that the use of infinity in scientific discussion was a kind of intellectual Godwin’s Law - if you have to invoke infinity to prove that something **can** happen, you are effectively proving that it won’t. I wish I could explain this better. At the time I found the argument really compelling and I have to admit I was happy to use infinite possibility as an argument in favour of lots of things. ---
**Author:** ![sealamp](https://avatars.discourse-cdn.com/v4/letter/s/3ec8ea/32.png) [@sealamp](https://boards.straightdope.com/u/sealamp)\ **Post date:** [November 20, 2003, 5:09am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/131 "2003-11-20T05:09:48Z")
This should resolve the whole discussion: [http://planetmath.org/encyclopedia/BorelCantelliLemma.html](http://planetmath.org/encyclopedia/BorelCantelliLemma.html) What else do you want to know? ---
**Author:** ![sealamp](https://avatars.discourse-cdn.com/v4/letter/s/3ec8ea/32.png) [@sealamp](https://boards.straightdope.com/u/sealamp)\ **Post date:** [November 20, 2003, 5:16am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/132 "2003-11-20T05:16:38Z")
This should resolve the whole discussion: [http://planetmath.org/encyclopedia/BorelCantelliLemma.html](http://planetmath.org/encyclopedia/BorelCantelliLemma.html) What else do you want to know? ---
**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 20, 2003, 11:46am UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/133 "2003-11-20T11:46:01Z")
Sounds good **don’t ask** - thats exactly the point i’m trying to make: infinity doesn’t have any bearing on whats probable, and arguments that use infinity to justify something that might be the case are fallacious. It doesn’t matter how probable it actually is in a finite universe, although people tend to use infinity to back up highly improbable situations. ---
**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 20, 2003, 4:10pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/134 "2003-11-20T16:10:12Z")
What would the probability of an identical mind to **Meta-Gumble** ’s in a Universe that is inite but very large. If the Universe was for instance so large that you can not express its width in light years using exponentials (10[sup]10[sup]…[sup]…[/sup][/sup][/sup] where it would require more 10’s to be {sup}ed on top of one another than we can express easily)? Would it not be incredibly likely that there would be huge numbers of **Meta-Gumble** (and Og save us **Bippy** ) minds out there? Can we avoid considering infinity at all by just considering a very large universe? ---
**Author:** ![Sergio](https://avatars.discourse-cdn.com/v4/letter/s/b77776/32.png) [@Sergio](https://boards.straightdope.com/u/Sergio)\ **Post date:** [November 20, 2003, 5:20pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/135 "2003-11-20T17:20:29Z")
**Bippy** Let’s forget the need of a identical mind and fogus only in the human genome. Our genome has 3 billion base pairs. Since there are four bases to form those pairs we have 4[/sup]3000000000[/sup] ---
**Author:** ![Sergio](https://avatars.discourse-cdn.com/v4/letter/s/b77776/32.png) [@Sergio](https://boards.straightdope.com/u/Sergio)\ **Post date:** [November 20, 2003, 5:28pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/136 "2003-11-20T17:28:29Z")
**Bippy** Let’s forget the need of an identical mind and focus only in the human genome. Our genome has 3 billion base pairs. Since there are four bases to form those pairs we have 4[sup]3000000000[/sup] possible different human beings. Since our Universe is believed to have in the order of 10[sup]80[/sup] particles, I can’t imagine how big should the Universe be in order to have a small probability to find another genetically identical Meta-Gumble. Of course, if this hypotetical Universe contained more than 4[sup]3000000000[/sup] human beings, some of them should have identical genomes, but this does not guarantee that one of the repeated persons would be Meta-Gumble. ---
**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 20, 2003, 6:23pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/137 "2003-11-20T18:23:09Z")
I’m considering here a universe much bigger than 10[sup]80[sup]80[sup]80[/sup][/sup][/sup] particles large, that is a Universe with incredibly flat curvature, but not so flat as to be infinitely large. Of course most of such a Universe is outside our 15 billion light year space like cone so most of such a universe is unable in any way to effect own position in the universe (assuming no faster than light travel is possible). ---
**Author:** ![Sergio](https://avatars.discourse-cdn.com/v4/letter/s/b77776/32.png) [@Sergio](https://boards.straightdope.com/u/Sergio)\ **Post date:** [November 20, 2003, 8:40pm UTC](https://boards.straightdope.com/t/infinity-and-probability/211684/138 "2003-11-20T20:40:26Z")
> [@](#): > > \*Originally posted by Bippy the Beardless \* > \*\*I’m considering here a universe much bigger than 10[sup]80[sup]80[sup]80[/sup][/sup][/sup] particles large, that is a Universe with incredibly flat curvature, but not so flat as to be infinitely large. Of course most of such a Universe is outside our 15 billion light year space like cone so most of such a universe is unable in any way to effect own position in the universe (assuming no faster than light travel is possible). \*\* It seems your Universe is not big enough for having any probability of a duplicate human being. [Previous page](https://boards.straightdope.com/t/infinity-and-probability/211684.md?page=6)