Name of this probability paradox?

Factual Questions
1

While thinking about the Sleeping Beauty paradox, I cooked up the following more-mundane seeming probability paradox. I’m sure that it must have a well-known name. It must be part of the standard collection that includes the birthday paradox, the boy or girl paradox, and the infamous Monty Hall problem. However, none of the pages in the Wikipedia category Probability theory paradoxes seems to match my paradox.

Here is my description of the paradox:

To my mind, this paradox is basically equivalent to the Sleeping Beauty paradox. But, if that were generally believed to be the case, the SB paradox wouldn’t have generated such a vast epistemology literature. My paradox would generate no controversy. It seems like the kind of thing that would be routine in beginning probability textbooks, just to show how seemingly-useless information can dramatically change probabilities. That’s why I expect that it must have a name.

2

I don’t have an answer to your question, but this bit needs to be a little better clarified. You may have selected 96 because it was on the (or a) cupcake you ate, or you may have selected the number 96 first, then told me whether you ate it.

3

Are you saying that each outcome (eating all 100 cupcakes or eating only the frosting-1 cupcake) necessarily entails eating a cupcake whose frosting-label and underside-label are the same? Because neither outcome does. Clearly, frosting-1 could have any label underneath it, so it you flip tails, 99% of the time, you will not eat a matched-label cupcake.

What if you flip heads and must eat all the cupcakes? Suppose the cupcakes were arranged in a single row, going from frosting-1 on the far left end to frosting-100 on the far right end sequentially? Is there a sequence n_1, n_2, …, n_100 such that n_i =/= i for all values of i?

Why yes! One candidate is 100, 99, 98, …, 1. Notice for all values of i, n_i + i=101. Were any of the labels identical, you would have n_i + i = i + i = 2*i =101. Since the labels i are integers, this cannot be the case.

Because I think your paradox turned on the premise that there is no Bayesian inference value to the information “I ate a matched-label cupcake” because that necessarily followed from your setup of the problem, the fact that such a claim does not follow would seem to vitiate the force of the supposed paradox. But perhaps not?

4

It sounds more like Monty Hall to me. I think you need to clarify the rules - if you had declared that, in the event that you only ate one cupcake, you would say the hidden number of that cupcake, otherwise you would randomly choose one of the numbers, then your declaration of “96” or whatever would not make it any more likely that you had eaten all the cupcakes.

If you had declared that you would say “96” if you happened to eat that particular cupcake, that is quite a different situation, and I don’t see the paradox.

5

Actually, it’s the same as the two-children paradox. If I ask you, “Did you eat the cupcake with bottom number 97?”, and you say yes, then that’s pretty strong evidence that you probably ate all of them. But if you select one cupcake at random from the set you ate, and tell me that number, and the number happens to be 97, then that doesn’t tell me anything.

6

If I gave you more information, it would change the probabilities. So I can’t “clarify” that point without changing the problem. (Or do I misunderstand you?)

[QUOTE=Kimmy_Gibbler]

Are you saying that each outcome (eating all 100 cupcakes or eating only the frosting-1 cupcake) necessarily entails eating a cupcake whose frosting-label and underside-label are the same?
[/QUOTE]

No, what I meant was so tautologous that it was probably just confusing to make it explicit. I only meant this: The underside label of each cupcake is equal to that very same underside label. In particular, this is true of some cupcake that I ate.

[QUOTE=Ximenean]
It sounds more like Monty Hall to me. I think you need to clarify the rules - if you had declared that, in the event that you only ate one cupcake, you would say the hidden number of that cupcake,
[/quote]

I did intend for this to be the case. If I ate only one cupcake, then 96 was the hidden label for that cupcake.

My declaration of 96 was random in the sense that you know absolutely nothing about it other than that it was the hidden label of one of the (possibly 1 or possibly 100) cupcakes that I ate. In particular, you don’t know how I selected that particular number from among the cupcakes that I ate. You don’t know whether I used a random number generator, or whether instead I had some rule like “give the lowest underside label”. You only know that the number I gave was among the eaten.

But even when this is all you know, it is in fact more likely that I ate all the cupcakes. I’ll give the computation in my next post.

7

[QUOTE=Chronos]
But if you select one cupcake at random from the set you ate, and tell me that number, and the number happens to be 97, then that doesn’t tell me anything.
[/QUOTE]

I’m pretty sure that that’s wrong. Unless I’ve made a mistake myself, you’ve just given the seductive reasoning for the wrong conclusion. If I hadn’t told you that number, the probability that I ate all cupcakes would only have been 50%. But once I give you a specific number, the probability becomes >99%. That’s the paradox.

Here’s the computation:

Let p(P) denote the prior probability of a proposition P, where “prior” means “before I told you any underside labels of any cupcakes that I ate.”

Let U be the proposition that I ate only one cupcake, so that

p(U) = 1/2.

Let A be the proposition that I ate all the cupcakes, so that

p(A) = 1/2.

Let E[sub]96[/sub] be the proposition that I ate the cupcake with underside label 96. Hence, the prior probability of E[sub]96[/sub] is given by

p(E[sub]96[/sub])

= p(E[sub]96[/sub]|U) * p(U) + p(E[sub]96[/sub]|A) * p(A)

= 1/100 * 1/2 + 1 * 1/2

= 101/200.

Now I tell you that I ate the cupcake with underside label 96. The problem is now to compute the probability that I ate all the cupcakes given this new information–that is, to compute the conditional probability

p(A|E[sub]96[/sub]).

Using Bayes’s formula, one computes

p(A|E[sub]96[/sub]) = p(A) * p(E[sub]96[/sub]|A) / p(E[sub]96[/sub]).

Clearly, if I had eaten all the cupcakes, then I would have in particular eaten the cupcake with underside label 96. Hence, p(E[sub]96[/sub]|A) = 1. Therefore,

p(A|E[sub]96[/sub])

= p(A) / p(E[sub]96[/sub])

= (1/2) / (101/200)

= 100/101 > 99%.

8

Clearly, there’s an error in your proof, because you can always tell me the number on the bottom of one of the cupcakes you ate, but you’re not eating all the cupcakes 99 percent of the time.

I think the error is that you haven’t included the probability of selecting 96 in the case where you’ve eaten all the cupcakes. You might have told us any number. In the single-cupcake-eaten scenario, you’re forced to tell us the number under that one cupcake.

9

Can’t answer any of the meaningful questions. But this:

In other words, of the cupcakes you ate, at least one of them had a number on the bottom? You’re correct - that tells you exactly nothing, since all the cupcakes had some number on the bottom.

10

ZenBeam, I agree that the result is counter-intuitive for the reason you say. But can you tell me exactly which line of my proof fails to follow from what precedes it? Better yet, can you provide an alternative computation for what you believe is the correct answer?

11

No, it’s not an error. It’s the distinction between “There is some specific label such that I am conditioning on the information of that label having been eaten” and “I am conditioning on the information of there being some label such that that label has been eaten”.

ETA: That was in reply to ZenBeam.

12

Irishman, right, you already know that any given eaten cupcake had some number on the bottom. That’s not new information, so making it explicit doesn’t change the original probability of 1/2 that I ate all the cupcakes.

Nonetheless, and this is the paradox, once you know that 96 specifically is one of those numbers, the probability changes dramatically. This is even though you wouldn’t know cupcake 96 from Adam.

Just to be clear if it wasn’t already: In the story, 96 is just a number that I volunteered to you as one of the cupcakes that I ate. You have no idea why I chose to give that number rather than any of the others, if there were any others. For example, you didn’t ask whether I ate cupcake 96 (as one of the other posters suggested). You were just informed of it.

13

Think of it this way: what seems paradoxical is that conditioning on the information “Label X was eaten” should shift the probability of “All 100 were eaten” up by the same significant amount, no matter what X is.

One feels the impulse to reply: But if it doesn’t matter what X is, my current unconditioned probability should already be just as high as this shift would take it to! Why wait for the inevitable before updating?

But the error of this supposed objection is: This reasoning only applies if, of all the possible values for X, they are already known to be exhaustive and mutually exclusive [so that the current probability will be the weighted average of each such conditioned probability, weighted by the probability of the event corresponding to each label]. But, in this case, they aren’t known to be exclusive; it very much could be the case that more than one label was eaten. Thus breaking the reasoning of this objection.

ETA: Ok, and that wasn’t in reply to Tyrrell; it was just an addendum to my previous post.

14

(Wherever I’m using “label”, I mean “underside-label”, though I think the “under-side label”/“over-side label” distinction was a bit of an unnecessarily confusing way to set the problem up. I would just have one set of labels, and have the setup be “Either I pick one cupcake at random, or I take all 100”, without worrying about distinguishing]

15

You might be right, Indistinguishable. I did have a reason for giving two systems of labels: I wanted it to be clear that the reader is supposed to be able to distinguish these cupcakes. That is, the reader has an objective means of picking out one cupcake explicitly from all the rest, without needing to use knowledge available only to the cupcake-eater.

But maybe it’s not so important to emphasize that for this problem. My desire to stress it probably comes from how I came to this scenario by thinking about the Sleeping Beauty paradox.

16

The error in your proof is that the act of telling me you ate cupcake 96 is different from the act of eating cupcake 96.

In other words, I need to use the prior for you telling me what you did.

Call it tE[sub]96[/sub]:

p(tE[sub]96[/sub]) = p(tE[sub]96[/sub]|U) * p(U) + p(tE[sub]96[/sub]|A) * p(A)

= 1 * 1/2 + 1 * 1/2

Here I’ve assumed that under U you would have told me which cupcake you ate regardless of which one it was. So it might be better labeled tE[sub]c[/sub], where c is a randomly selected cupcake from C, the set of cupcakes you ate. In U, C has only one member, and in A, none of the members are different from each other. In other words, c doesn’t give as any new information.

This is much like the Monty Hall and the two-child problem, where how you discover the information can have an effect on probability. I think that is, for some reason, counter-intuitive since many people don’t expect probability to have an epistemological bias.

17

It’s similar to the version of the Monty Hall problem where Monty is allowed choose whether or not to open an empty door. If Monty decides to only open doors when your initial pick was correct, your chances of winning by switching doors are always 0%. If Monty decides to only opens doors randomly, your chances of winning are now 66% if you switch (once he opens a door).

But if this is the first time the game is played (and Monty proceeds to open a door), there isn’t enough historical data to know whether Monty’s intentionally gaming the system or not, so you have no clue what your chances are by switching.

18

Well, if you do particularly want that, that’s alright. It’s just having two notions of “label” which strikes me as distracting; instead, we can say there’s one blue cupcake and the rest are normal-colored.

ETA: I will never get a reply in at the right time in this thread…

19

That’s often a valid distinction to make, for the way in which many probability questions/riddles are set up, but it is still legitimate to speak of conditioning on information which isn’t of the form “Person Y tells me fact X [which is also true]”. One can just condition on fact X directly; whether this is what the questioner is actually asking about or not is another matter (the problem with so much probabilistic paradox is that the ordinary language of probability leaves so much implicit and to be reconstructed by convention, though often intuitive reconstruction (of what was being asked, which is not a math problem but a language problem) will be different from what the questioner, at the gotcha moment, demands). In this problem, it’s clear that we are being asked to directly condition on a fact about a particular label being eaten, without worrying about any intermediary and the fact that they brought this to our attention.

Though, in a way, if we were to worry about an intermediary who chooses which label to tell us about, this would simply be fixing the “Well, the possibilities for which label was eaten aren’t exhaustive and mutually exclusive” mistake in the objection from before, since, presumably, the possibilities for which label we’re told was eaten will be exclusive.

20

For the purposes of this problem, you may assume that I’m completely honest. My saying that I ate cupcake 96 is as good as my having actually done so.

It is not correct to say that p(tE[sub]96[/sub]|U) = 1.

Consider the time before I’ve told you a label of a cupcake I’ve eaten. Suppose that, at that time, you learned somehow that I ate only one cupcake, though you don’t know which. What is the probability that I’m going to tell you that I ate cupcake 96? That probability is, by definition, p(tE[sub]96[/sub]|U). It can’t be 1, because that would mean that you could predict which cupcake I ate given only the knowledge that I ate only one of them.