# if probability = 1; means event must occur ? **URL:** \ **Category:** Factual Questions\ **Created:** [August 27, 2002, 8:22pm UTC](https://boards.straightdope.com/t/if-probability-1-means-event-must-occur/125827 "2002-08-27T20:22:48Z")\ **Posts on this page:** 3\ **Page:** 3
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [August 29, 2002, 4:43am UTC](https://boards.straightdope.com/t/if-probability-1-means-event-must-occur/125827/41 "2002-08-29T04:43:31Z")
About the axiom of choice, what I was trying to say is that I’d be willing to bet, if you polled mathematicians on whether it is true or not, a majority of them would say that it is true; it has become a pretty standard axiom in set theory. Mathematics can get a lot uglier without the axiom of choice: Given two sets A and B, you cannot compare the cardinalities A and B. In fact, the cardinality of A (or B) is not even guaranteed to exist. Given an infinite set, it may not have a countably infinite subset. You can’t talk about arbitrary Cartesian products of sets. Some vector spaces won’t have bases. About the randomA vs. randomB, I agree with **ultrafilter** ’s earlier answer. ---
**Author:** ![Race\_Bannon](https://avatars.discourse-cdn.com/v4/letter/r/5f8ce5/32.png) [@Race\_Bannon](https://boards.straightdope.com/u/Race_Bannon)\ **Post date:** [August 30, 2002, 3:40am UTC](https://boards.straightdope.com/t/if-probability-1-means-event-must-occur/125827/42 "2002-08-30T03:40:24Z")
> [@](#): > > \*Originally posted by Cabbage \* > \*\* > The probability for any _particular_ number to be picked is zero. However, we know _some_ number b will be picked, so here we have an example of an event with probability zero that actually does happen. Correspondingly, the probability that b will not be picked is one, and that _didn’t_ happen. \*\* I’m going to jump in late (and probably over my head) but this is what’s bothering me. You claim that the Lebesgue measure is a legitimate probibility distribution, and that P(something)=1 is not necessarily GTH. I don’t understand how you claim that something gets picked. That is, if P(b)=0 for all b in [0,1], I could claim that no number ever gets picked at all. That would seem to violate the probability axiom P(S)=1, where S=[0,1], but there you have it. That is, I think you’re claiming that P(S)=1 means that _something_ gets picked, but the probibility that that something gets picked is zero. Seems like you could save a step by saying that P(b)=0 means nothing ever gets picked, therefore P(S)=1 doesn’t mean that probibility =1 means GTH. Now, getting to my problem with this - either you have to say that P()=1 means GTH, or P()=0 means impossible. You can’t have it both ways, can you? ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [August 30, 2002, 2:06pm UTC](https://boards.straightdope.com/t/if-probability-1-means-event-must-occur/125827/43 "2002-08-30T14:06:48Z")
All right, let me try a slightly different, more concrete approach. Say your cable goes out, so you call the cable company for a repair man. They tell you they’ll send somebody out between noon and 1PM. You ask, “Could you be any more specific?”, to which they reply, “I’m sorry, I can guarantee that he’ll be there between noon and 1PM, but other than that, I really have no idea what time to expect him.” How can we model the probability of him arriving at a certain time? We know it’ll be between noon and 1PM, but, other than that, no time is preferred over any other. So we can model this analogously to my previous example–picking a real number at random from [0,1] (corresponding to the number of hours after noon), only now we don’t have to worry about issues of how we actually pick the number–the cable guy simply arrives when he arrives. As before, the probability of him arriving at any particular time is zero; still, he will arrive at _some_ particular time. Now I know that some may object to this model, possibly by saying “Time isn’t continuous,” or something to that effect. However, we use, for example, continuous functions in calculus all the time to model the real world; how can you really object to this model? **kellymccauley** : > [@](#): > > That is, if P(b)=0 for all b in [0,1], I could claim that no number ever gets picked at all. That would seem to violate the probability axiom P(S)=1, where S=[0,1], but there you have it. A probability function is required to be countably additive, but here the sample space is uncountable. It doesn’t violate the probability axioms to have a probability function that’s zero at every point, yet one on the whole sample space. (In other words, P(a single point)=0, while P(whole sample space)=1). > [@](#): > > That is, I think you’re claiming that P(S)=1 means that something gets picked, but the probibility that that something gets picked is zero. I’m claiming something gets picked, but not because P(S)=1. I’m claiming that something gets picked because that’s given in the problem, just like the cable guy is guaranteed to show up between 12 and 1. The probability any _particular_ thing is picked, but, _taken as a whole_, we know _something_ must get picked. > [@](#): > > Seems like you could save a step by saying that P(b)=0 means nothing ever gets picked, therefore P(S)=1 doesn’t mean that probibility =1 means GTH. I don’t follow this. > [@](#): > > Now, getting to my problem with this - either you have to say that P()=1 means GTH, or P()=0 means impossible. You can’t have it both ways, can you? Sure I can, why not? Think of it this way: Things with probability zero come in two categories: “impossible” and “damn near impossible”. Things with probability one also come in two categories: “certain” and “damn near certain”. I see no contradiction. [Previous page](https://boards.straightdope.com/t/if-probability-1-means-event-must-occur/125827.md?page=2)