How To Resolve All Probability Paradoxes: Apples In Sack Example

Via Alexander Bogomolny?:

At 1 minute to midnight 10 apples fall into a sack. The same happens at half a minute to midnight, then at a quarter minute to midnight, and so on. At each such event you remove an apple randomly from the ones still present in the sack.

What is the probability that at the midnight strike the sack will be empty?

You’re supposed to say “1”. I would say 0.

I’d be just as wrong as the folks who say 1. In the reason we’d all be wrong lies the answer to solving all probability paradoxes.

Here is all we need remember to solve all probability paradoxes: all probability is conditional.

The conditions include those explicitly stated, such as in the words and numbers of the premises to some problem, and those implied, such as what the words mean. It cannot be any surprise that two people could look at the same passage and come away with two different implied meanings of that one passage. How many interpretations are there of any one Shakespeare sonnet?

The common error is to assume passages need no implicit premises, that, somehow, passages when directed toward math are immune from multiple interpretations. This is, of course, false. But mathematicians try to make it true, but being as careful as possible, and, when they can, lapsing into pure symbolism, where every stroke has an explicitly defined meaning. In that case, there is only one right answer.

That’s no so in the apple example. There are many ambiguities. The trick, then, is to recognize them, define them, then solve the probability for them, while acknowledging that different resolutions can give different answers.

If two people come to a different understanding of what the words, to include the grammar, of a passage means, these people could come to a different probability. Assuming no mistakes in calculation—a large assumption for complex passages—then each person would give correct but different problems.

Equally, each person would give the wrong probability—if they were not explicitly clear about their implicit assumptions.

Here’s my take, i.e. my list of implied premises, for the apple problem. All “events” must take place in finite time. However long it takes, it must take some finite time to put apples in and take one out. This time does not have to be constant, even; though I believe the words of the problem imply that it is.

Regardless of the length of this discrete event, there will come a time when the clock strikes midnight. Because we could only add a finite amount of apples from 11:59 PM to 12:00 Midnight, and so we could only take out a finite amount, each event, adding and subtracting, takes a discrete tick of the clock, and there are only a finite number of clicks. And since we take only one—and there is no way to take any apple “randomly”: one must take an apple some way—for every ten added, the sack is guaranteed to have apples in it at midnight. Therefore, the probability the sack is empty is 0.

Further, my implied premises hold (I say) for any event happening in physical (contingent) reality, by which I mean in physical substances. There is no real physical or energetic process that can take an infinite number of steps in a finite time. So whether it’s apples or quarks or strings or whatever, the probability will always be 0.

But physical reality is not all there is to reality. There is also spiritual, or intellectual reality where the restrictions on time do not apply. With that in mind, here are the list of implied premises for those who answer 1, i.e. the probability almost surely (in math terms) that the sack will be empty.

There are, in this second interpretation, an infinite number of steps at which apples will be added, and an equal number of infinite steps where apples will be removed. In short, a countably infinite number of apples will be added, and a countably infinite number of apples will be removed. Mathematicians will understand that because both sequences are countably infinite, they can be “lined up” against each other in a one-to-one correspondence. (This is the same reason why all fractions of the type p/q, where p and q are integers, are the same “size” as the number of integers.)

Because the cardinality (the size) of both adding and subtracting apples are equal, and because we are in the strange land of Infinity, and therefore because the number of apples added can be put in one-to-one list of apples removed, the probability the sack is empty goes to 1.

That is the right answer to the second set of implied premises. Just as 0 is the right answer to the first set of implied premises. Again, since all probability is conditional on the premises assumed, both answers are correct, but only after the implied premises have been made explicit.

Both sets of implications are possible from English in the passage. Are there other possible implied premises? Maybe. I don’t see any that would change the probability. But this is a relatively simple passage. Others, notoriously, are not. Heated disagreements occur when one set of folks say “It’s obvious these implications hold” while others say, “No, these are!”


  1. PK

    If nine apples are effectively added to the bag in each time increment, the why doesn’t the number of apples in the bad increase without bound? It will never be empty.

    Or, as the title of the post implies, is there more than one bag? A new bag of 10 apples each time?

  2. JohnK

    Along with being a concise, even a brilliant, example of the absolute necessity to explicate one’s premises as part of any sensible deduction and calculation of probability, this brief piece amounts to a ‘toy proof’ (given Matt’s premises), that, at very least, anyone calculating the probability of any physical event – ever – should exhaust every possibility of making a discrete calculation – a calculation therefore firmly founded in reality – rather than resorting to infinities.

    (I opine that some Old World statistics, with their infinities, parameters, and downright fudging, were invented because, at the time, making the precise discrete calculations would have been an unimaginably long process. Nowadays many of these calculations are trivial, of course – but perhaps not all. Hence my own fudging about possible careful resort to infinities in some possible circumstances, even today).

    However, I continue to insist that Matt makes a category error when he conflates, or confuses, ‘intellectual’ reality, with spiritual reality. Sometimes I like to say, “I’m not spiritual – I’m Catholic,” to hit at the fundamentally sacramental nature of Reality, a Reality categorically beyond the ken of Aristotle and Plato, but proved in the crucible of Christ Jesus, Crucified and Risen, bridegroom of His Church.

  3. Sheri

    PK: I read it as being one bag, with 10 apples added each time. I do see your point—how can adding 10 and subtracting one ever even out. I don’t think it can, even with adding versus subtracting as a one-to-one event, you cannot reach zero.

  4. Briggs


    One bag. Ignore title.

  5. D. Steven Fraser

    The next assumption is the time required for the apples to dump each time. if instantaneous, then the problem becomes infinite. If the dump takes a finite time (1-tick, or any other constant interval) then the cycle is not infinite.

    The same is true for the time of the removal. If _either_ one takes the same finite time, then the problem can be expressed in the relationship between the longer of the two times, and 1 second. This also controls how many dumps and removals there have been in the minute.

    There is narrow set of constraints based on what is actually mentioned as occurring: that the apples dump at the times described, and that 1 apple is removed. If the time allotted for the removal can be reasonably assumed to take place in the interval between one dump occurrence and the next, we can consider that pair of things as a single, multi-stage event. This scenario is an infinite one.

    In this infinite scenario, the real-world clock never gets to 12:00, so the bag is never checked, and the probability of emptiness cannot be determined.

  6. Ye Olde Statistician

    In this infinite scenario, the real-world clock never gets to 12:00, so the bag is never checked, and the probability of emptiness cannot be determined.

    Or when you do open the bag, you discover the cat is alive, has eaten all the apples, and you have let the cat out of the bag.

  7. Nate

    “Infinity is a dreadfully poor place. They can never manage to make ends meet.” – The children’s story ‘The Phantom Tollbooth.’

  8. JohnK

    I beg forgiveness for this one-time OT comment, but since Matt has frequently written about consciousness, the brain, ‘intellection’, etc., and many commenters also seem interested in these subjects, here is a belated link to a May 18, 2016 piece by senior research psychologist Robert Epstein, titled “The empty brain: Your brain does not process information, retrieve knowledge or store memories. In short: your brain is not a computer.”

    In the piece, Dr. Epstein almost just ‘points out’ – his argument is that simple and elegant – that the current Standard Model of how the brain works is complete rubbish, obviously and provably false, and that this has immediate implications; for instance, that we have wasted billions on ‘science’ founded in a metaphor that is abjectly silly, and that “we will never have to worry about a human mind going amok in cyberspace.”

    I also noted with some amusement a certain similarity to Matt’s current intellectual plight; to whit, to get almost no traction, even though you’re provably correct: “the mainstream cognitive sciences continue to wallow uncritically in the IP [Information Processing] metaphor…” even though “the IP metaphor is not even slightly valid.”

    Anyway – see what you think.

  9. Sander van der Wal

    Sacks (and bags) have a finite capacity for holding apples. Earth itself has a finite capacity for making apples. For the apples to get into the sack very close to to midnight, at some point they will need to move faster than light to get into the sack before the next bunch comes along.

    So that’s the secret. They move so fast there won’t be a whole apple in the sack, just apple sauce.

  10. Andrew White

    Tof has it right. Mathematically speaking, the problem is undefined at midnight, although the limit is trivially calculable and it’s not 0

    Note that this example differs from the Tortoise “paradox” in that the latter is physically sensible (it’s our way of modeling the situation that is distorted, not the situation itself)

  11. Thank you, John K. That’s a fine article. By the way, Roger Penrose has said much the same thing, although in a mathematical context: consciousness is not to be achieved by algorithmic processes. See “The Emperor’s New Mind” and books following that.

  12. Ken buddy- what has happened?!? WTF??? You used to have at least a slightly interesting blog and then retreated over the last 6-9 months into an incredible shield of boring games and stats?

    Why so glum?

  13. Jim Fedako

    PK and Sheri —

    I agree. The irony is the link provided by Ken includes tweets by Taleb, the very same man who laughed off such “angels dancing on a pin” nonsense in one of his books .

    This is all a result of folks assuming mathematics is reality and not a model of reality, ignoring that mathematics is either incomplete or inconsistent, or both — a reification error.

  14. Joy

    YOS Roger Penrose was there first.

  15. Ken

    Dixon – Been busy dealing with real-world nonsense…I suppose its a yin-yang sort of thing … having to deal with and straighten out truly significant matters offsets blog-related playtime. In other words, when my BS meter pegs out I shift focus to where it matters most …

  16. Ye Olde Statistician

    1961 et seq. Lucas’ Goedelian proofs.
    1969. Stanley L. Jaki, Brain, Mind and Computers
    1980/1984. John Searle’s Chinese room and elaborations
    1989. Penrose, The Emperor’s New Mind: Concerning Computers, Minds

  17. Joy

    Which point in the mind computer argument are you making?
    That thought is not only computational?
    Penrose has some other ideas about thought and gives ideas about where and how it might happen.
    I still think the mind is not reducible to matter and energy.
    It doesn’t impinge that some think it is. The truth is a great limiting factor.

  18. grodrigues


    “Which point in the mind computer argument are you making?”

    I am not YOS, but honestly, isn’t it obvious the point he is making? You said and I quote “YOS Roger Penrose was there first.” YOS gave the dates in response to the quote. Do the math.

  19. šiki?

    10-1+10-1+10-1+10-1+ … = infinity

    10-1-1-1-1-1-1-1-1-1+10-1-1- … = 0

    the fact is that infinite addition is not commutative, if you suppose it is you can get any integer and +/- infinity as the “result”

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