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# The Balls in the Urn Paradox (Ross–Littlewood)

This paradox was originally described by mathematician John Littlewood in his 1953 book Littlewood’s Miscellany, (Footnote: John Edensor Littlewood, A mathematician’s miscellany, London Methuen, 1953.) and was also included by Sheldon Ross in his 1988 book A First Course in Probability. (Footnote: Sheldon Ross, A First Course in Probability 8th Edition. Pearson, 2009 )

The question as posed in Ross’s book is:

Suppose that there is an infinitely large urn and an infinite collection of balls each labeled with a unique number, 1, 2, 3, … . Consider the following process:

At 60 seconds before noon, balls numbered 1 through 10 are placed in the urn and ball number 10 is withdrawn. (we assume that the withdrawal takes no time.) At 30 seconds before noon, balls numbered 11 through 20 are placed in the urn and ball number 20 is withdrawn. At 15 seconds before noon, balls numbered 21 through 30 are placed in the urn and ball number 30 is withdrawn. And so on, and so on…

The question is: How many balls are in the urn at noon?

Ross says that the answer is clear, and is what most people would say - there is an infinite number of balls in the urn. This is in accordance with most people’s intuition, based on the fact that at each step more balls are added than are being removed.

## Muddying the waters: Round 1

However, the question is now extended to the situation where at step 1, balls 1 through 10 are inserted into the vase, and ball number 1 is then removed. And at step 2, balls 11 through 20 are inserted, and ball 2 is then removed. And so on.

According to Ross & Littlewood, every ball labeled n that is put into the vase is eventually removed in a subsequent step, the step n, and hence, by noon the vase must be is empty.

It’s easy to see the fallacy. (Footnote: Whether the balls are numbered or not is irrelevant. One could just as easily set the balls out in a line when putting the balls into the urn (the urn is infinitely large anyway) and simply remove according to the position on the line. Then the balls are completely identical, and it now means that the claim is now that it makes a difference to the result depending on where in the line you remove the ball from!) The process is an infinite unending process that is not completed (the reference to time in the description is a red herring, see Time as a red herring below). Because it is an infinite unending process, to make a calculation requires applying a limiting condition. Since the value in the problem which does not reach a finite endpoint value is n, then the limiting condition must be a condition which applies as n → ∞.

So what value are we going to calculate as n → ∞? Well, we can calculate the number of balls remaining in the urn as n → ∞. It’s obvious that after each step n (regardless of which ball is actually removed at that step) there are 9n balls remaining in the vase. And the limit of 9n as n → ∞ is infinite, or it might be better to say that there is no upper limit to the value of 9n. Hence there is no upper limit to the number of balls in the urn as the process continues indefinitely.

And we can also calculate the number of balls removed as n → ∞. It’s absurdly simple - as n → ∞ the number of balls removed approaches infinity. So the number of balls remaining in the urn and the number of balls removed, in the limit, become infinite.

### The Duplicate Urn

Another way to see the fallacy in Ross’s argument is given by the following;

Consider the case where there is another urn B.

After each step more balls are added to urn B so that it has the same number of balls as the original urn. We note that at every step more balls are put into the original urn than are removed from it, so at every step more balls are added to urn B.

So after the first step, urn B has 9 balls, after the next step it has 18 balls, and after the next step it has 27 balls, and so on. Each time urn B has 9 more balls than at the previous step.

So, what does the urn B contain at noon?

It can’t be an empty urn, since no balls are ever removed from urn B. And if the urn B can't be an empty urn, neither can the original one.

## Muddying the waters: Time as a red herring

It is a fallacy to believe that the definition means that the “time” noon is reached. But the process, by definition, never completes - there are infinitely many steps.

So ask the question: Does the definition define that noon is reached?

Answer: No, it doesn’t.

Part of the problem is that people think of the word “time” where it occurs in the problem as though it is real physical time, but where there is no limit to how small it can be sub-divided. But in the problem time is simply a mathematical dimension, and if one thinks of it in the same way as a physical length, then one can see that the “time noon” is never reached by the process - the definition only defines that it comes closer and closer without being reached.

Asking how many balls will be left at noon invites one to assume that there is no contradiction between the definition given and the assumption that noon can be reached by the definition.

The paradox is a result of an inherent contradiction in the question that is asked. The fact that most people answer that there will be infinitely many balls in the urn is simply because they assume that the poser of the question intended it to be meaningful, and this response is the best way of dealing with the contradiction that results from that assumption.

## Muddying the waters: Round 2

In his book, Ross muddies the waters even further by considering if the choice of which ball removed at each step is completely random, and calculating the probability of a ball of number n having being removed by some step m. Obviously, as the number of steps increases, the probability of that particular ball being removed, on or before that step, becomes ever greater. The limit of this probability is a probability of 1. But that is a limit that is not actually reached by the unending infinite process.

On the other hand, for any ball, the probability that there is some step n at which it is removed is, of course, 1 - it is a certainty. And the converse, the probability that there is no step n at which it is removed is 0. It takes Ross several equations to come to this outstandingly obvious conclusion.

Ross takes that probability of 0 for one ball, and applies it to all the balls, and comes up with a result that the total probability that for every ball, there is no step at which it is not removed is also 0. Again an outstandingly obvious result.

The paradox only arises when Ross takes this simple statement and twists the meaning to read:

“The probability that the urn is not empty at 12 PM is 0.”

Question:

Where does Ross rake up this conclusion from? Where in his equations is there any mention of time, or 12 PM or noon?

Nowhere in his equations is there any mention of time in any form whatsoever, see the Appendix below. All they refer to is the probability of a ball being removed at some step in a limitless recursive process that never terminates. Ross’s equations are simply a sham facade that have absolutely nothing to do with Ross’s promotion of a Platonist agenda that an infinite process can in some way “finish”. He might as well have said:

“The probability that the urn is not empty when the limitlessly recursive process finishes is 0.”

It is interesting that there is a substantial discussion of the paradox at StackExchange: Balls in an urn where there is intense debate about the equations - there is one respondent who hits the nail on the head, see What’s the best explanation we can give to them to solve these conflicting intuitions? It is also interesting that it has been given hardly any up-votes, the voters preferring instead to wallow in Platonist fantasy land. These Platonists mock other people if they prefer their own intuition over the pseudo-mathematical arguments that these Platonists present, but the irony is that the Platonists’ own arguments involve their intuitive belief that the infinite process terminates.

You could just as easily ask the probability that, for any ball of number n, there will be balls left in the urn that have a number either less or greater than n - for which the probability is 1; and for any ball of number n, there will be no balls left in the urn that have a number either less or greater than n is a probability of 0. Applying that to all balls gives the opposite result to the above result.

#### And more…

For another example of a contradiction arising from the notion that an infinitely recursive process can somehow “complete”, see The Platonist Rod paradox. Footnotes: ### Appendix: Sheldon Ross’s equations

The following is from Example 6a, Probability and a paradox in Chapter 2 Axioms of Probability, in Sheldon Ross’s book A First Course in Probability 8th Edition. Pearson, 2009.

Define En to be the event that ball number 1 is still in the urn after the first n withdrawals have been made. Clearly [To understand this equation, just note that if ball number 1 is still to be in the urn after the first n withdrawals, the first ball withdrawn can be any one of 9, the second any one of 18 (there are 19 balls in the urn at the time of the second withdrawal, one of which must be ball number 1), and so on. The denominator is similarly obtained.]

Now, the event that ball number 1 is in the urn at 12 PM (Footnote: In fact these symbols makes no reference whatsoever to 12 PM, nor to time in any form. What they actually refer to is the totality of events where n is not the step where the ball 1 is removed. That’s all.) is just the event Because the events En, n ≥ 1, are decreasing events, it follows from Proposition 6.1 that

P{ball number 1 is in the urn at 12 PM} We now show that Since this is equivalent to showing that Now, for all m ≥ 1, Hence, letting n → ∞ and using the fact that yields Thus, letting Fi denote the event that ball number i is in the urn at 12 PM, (Footnote: As previously noted, in fact Fi makes no reference whatsoever to 12 PM, nor to time in any form. What they actually refer to is the totality of events where there is not a step n where the ball i is removed.) we have shown that P(F1) = 0. Similarly, we can show that P(Fi ) = 0 for all i.

(For instance, the same reasoning shows that for i = 11, 12, … , 20.)

Therefore, the probability that the urn is not empty at 12 PM (Footnote: Again, we must note that these equations make no reference whatsoever to 12 PM, nor to time in any form. What they actually refer to is the total probability of events where there is not a step n where the ball i is removed.) satisfies by Boole’s inequality. Thus, with probability 1, the urn will be empty at 12 PM. (Footnote: Yet again, it must be stressed that the above equations make no reference whatsoever to 12 PM, nor to time in any form.) Footnotes: Diverse opinions and criticisms are welcome, but messages that are frivolous, irrelevant or devoid of logical basis will be blocked. Difficulties in understanding the site content are usually best addressed by contacting me by e-mail. Note: you will be asked to provide an e-mail address - any address will do, it does not require verification. Your e-mail will only be used to notify you of replies to your comments - it will never be used for any other purpose and will not be displayed. If you cannot see any comments below, see Why isn’t the comment box loading?. ## NEWS

### Paper on the diagonal proof

There is now a paper that deals with the matter of language and the diagonal proof, see On Considerations of Language in the Diagonal Proof. ### Other recently added pages

The Myths of Platonism

The Platonist Rod paradox

The Balls in the Urn Paradox ### Lebesgue Measure

There is now a new page on a contradiction in Lebesgue measure theory. ### Easy Footnotes

I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly). ### O’Connor’s “computer checked” proof

I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.

### Previous Blog Posts

For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem

– a page with general links:

– and a page relating specifically to the Gödel mind-machine debate:

Gödel, Minds, and Machines

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