Consider a fair infinite lottery with tickets numbered ..., −3, −2, −1, 0, 1, 2, 3, ..... Consider these events:

*E*: winner is even*O*: winner is odd*E*^{*}: winner is even but not zero*E*^{+}: winner is even and positive*O*^{+}: winner is odd and positive*E*^{−}: winner is even and negative*O*^{−}: winner is odd and negative.

Plausibly:

*O*^{+}is equally likely as*O*^{−}*E*^{+}is equally likely as*E*^{−}*E*is equally likely as*O*all tickets are equally likely to win.

Now then *E*^{*} is less likely than *E* by one ticket, and hence also less likely than *O* by one ticket according to (3). And *E*^{*} is the same event as the disjunction *E*^{+} or *E*^{−}, while *O* is the same event as the disjunction *O*^{+} or *O*^{−}. Therefore, the disjunction *O*^{+} or *O*^{−} is one ticket more likely than the disjunction *E*^{+} or *E*^{−}. Since *O*^{+} and *O*^{−} are equally likely and *E*^{+} and *E*^{−} are equally likely, it follows that:

*E*^{+}is*half*a ticket less likely than*O*^{+}.

But how could one lottery outcome be less likely than another by *half* a ticket in a lottery where all tickets are equally likely to win? The only option seems to be that the probability of any particular ticket winning is zero. And that seems paradoxical, too.

## 16 comments:

The probability of any particular ticket winning is at most infinitesimally far from zero, because there are infinitely many tickets, and they are equally likely.

So, if we take a probability to be a real number, it is zero. (Similarly, if we take a probability to be a percentage, then it is zero.) Why is that paradoxical?

Perhaps we could say there is a non-zero probability of each ticket winning, since one ticket has to win, but the probability would asymptotically approach zero and so wouldn't be a particular number. This makes it difficult to talk about one set of tickets being more likely to win than another.

You have just shown us how a lottery outcome could be less likely than another by half a ticket, in a lottery where all tickets are equally likely to win. Basically, it makes no sense to say that E* is less likely than E by one ticket, when E* and E are made true by infinitely many tickets. Or rather, how is it that you have not simply shown that it does not make sense, in that case?

Sorry, NA, I addressed my last comment to Alex.

Regarding your comment though, you seem to think that a zero probability of something means that it cannot happen (since you say that since one ticket has to win, there is a non-zero probability of it winning). I wonder where you get that idea from. I mean, normally it is true. But normally there are not infinitely many things in play.

What do you think a probability is? In ordinary English, there can be vaguely high and vaguely low probabilities, which are not numerical. In mathematics, probabilities are usually real numbers, but then, by definition, they do not apply to scenarios like Alex's. Some mathematicians think that they can be infinitesimals, and they might apply, but the mathematics is messy. So, there seem to be a range of options available.

Alex, if you take any two tickets from E, to get E** (say), then it is the case that as many tickets make E** true as make E true, because of an obvious one-to-one correspondence.

So, if E* was less likely than E by one ticket, then E* would be more likely than E** by one ticket, and hence more likely than E by one ticket. So, if it makes sense, then it is absurd. So, it does not make sense.

No worries, Martin, I assumed as much.

I'm not sure what exactly a probability is. It seems like a very primitive concept (like truth, knowledge, causation, etc.) that may not admit of detailed analysis in non-probabilistic terms. For instance, if I were to try to define a probability as the chance or likelihood of a certain outcome, that just raises the question of what a "chance" or "likelihood" is. So I'm not sure.

As for "zero probability" meaning that something can't happen, that seems intuitively true to me. To say something has zero probability of occurring appears to be another way of saying that it can't happen. It's more or less tautological. And this is the case whether we are operating with this or that notion of probability, whether we are speaking in technical or non-technical fashion, and whether we are dealing with infinitely many things or only one.

As I reflect more on your idea that we have a "range of options" available in understanding probability, I'm beginning to think the perplexity in Alex's thought experiment comes from the fact that (1) there are infinitely many objects in play, and (2) we intuitively operate with a quantitative notion of probability (eg, 1/16, 1/1,000). But I'm not sure these belong in the same conceptual space. The notion of infinity is so counterintuitive and unlike anything in our everyday experience that it seems to necessitate the use of strictly qualitative notions of probability, so as to avoid paradoxical outcomes such as in Alex's thought experiment, where one set of tickets is half a ticket more likely to win than another. If we abandon the assumption that probability is quantitative in this case, and instead say it is qualitative (eg, the odds of each ticket winning are "vanishingly small"), then much of the thought experiment, including talk of one set of tickets being half a ticket more likely to win than another, simply becomes nonsensical.

Alex:

Is this surprising? Doesn’t it just reflect the lack of countable additivity? For example, E* differs from E by one ticket, so their probabilities will differ by the probability of one ticket (because the sum is finite). But E+ and O+ differ by infinitely many tickets, so there need be no special relation between their probabilities.

Also, intuitively, E+ is in a sense half way between O+ and O+ - {1} (in that it is O+ shifted one position to the right rather than two.) So intuitively, it is at least not surprising if the same might apply to the probabilities. :-)

I didn't say that the zero probability option was paradoxical, but only that it seemed to be.

I do wonder why it seems paradoxical to you, though.

Because it seems that it is more likely that ticket #2 will win than that the ticket whose number is the square root of 2 will win (there not being such a ticket).

Thanks.

That simplifies my answer to this puzzle (thanks too for the puzzle; it is good practice for me to try to solve puzzles, I think).

Ticket #2 does not just seem more likely to win, it is more likely to win. I therefore presume that what you mean is that it seems that probability zero should mean impossibility. As that is what NA was saying, my reply to him applies here, I think. Basically, it is not actually a necessary part of what the word "probability" means, that zero probability implies impossibility.

To flesh that out, I would use the analogy with percentages, which are a lot like probabilities: they run between 0 and 1 in bits of one hundredth, and can be used for frequencies. If one was looking at one-in-a-million chances, and one used percentages, one might say that the percentage chance for a one-in-a-million chance was zero. That zero clearly does not mean impossibility. Still, one really should be using real number probabilities for such things. And similarly, when one is talking about an infinite lottery, one would be wrong to think of a zero probability as meaning impossibility.

Perhaps one could use infinitesimals. If they do not pan out mathematically, then non-mathematical sense could be given to them. Here my analogy would be this: When you have no data on some proposition, it can make sense to think of it as being about as likely as not. The reference class paradox shows that that is not the same as a 50% chance of it being true. Similarly, one could say that this was infinitesimal in a non-mathematical sense: not no chance, and closer to zero than to any other real number probability. I can see little difference between this and calling it a zero probability (much as it is zero percent), but this is probably a matter of taste.

Basically, I think that you are right: it does seem paradoxical. And it does seem so in a similar way to the half-ticket seeming paradoxical. Nevertheless, one is incorrect (because there is no sense to "more likely by one ticket") and the other is not. There is that significant difference, which solves this puzzle (I think).

Martin:

Can you *both* say that A is more likely than B, but A and B have the same probability? That seems like a contradiction. It seems to me that the bullet you have to bite is nastier than you think: you have to deny that ticket #2 is more likely to win than the ticket whose number is the square root of two. In other words, you have to deny that the possibility is more likely than the impossible.

Well, it can indeed seem like a contradiction.

But consider two girls, Ann and Bev, who have the same size (in shoes or clothes or hats). Nevertheless, Ann is slightly bigger than Bev. Clearly there is no contradiction there, and my earlier use of "probability" was just like that use of "size" (as my talk of percentages was intended to make plain).

Presumably you do not think that, by asserting that Ann and Bev have the same size, in some sense, I

haveto deny that Ann is bigger than Bev, and so I wonder, do you still think that Ihaveto deny that the possible is more likely than the impossible?I do think that there is

a sensein which A is more probable than B (just as there isa sensein which the size of Ann is greater than the size of Bev). But I distinguish between that and the mathematical modelling of it, whether with real number probabilities or whole number percentages.I therefore believe that your position is worse than you think. I assume that you (in common with many others) are confusing "more probable" and "greater probability". Are you not confusing two quite different kinds of things here?

Alex,

I meant to say "more probable" in the non-mathematical sense of more expected, and "greater probability" in some mathematical sense, such as a bigger real number.

Thanks again...

Alexander Pruss,

The most common rebuttal I hear to your messenger/reaper paradox is that it only fulfills the burden of proof that that specific situation cannot occur, but it does not fulfill the burden of proof that the past can't be infinite.

What are your thoughts?

I discuss objections like this at length in the book.

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