The Apples In A Sack Paradox: Solve It Or You Get No Pie Tomorrow

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. And in that reason we’d all be wrong lies the answer to solving all probability paradoxes.

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 used to state the paradox, and those implied, such as what the words and symbols mean. It cannot be any surprise that two people could look at the same passage and come away with two different implied meanings of the same passage. How many interpretations are there of any one Shakespeare sonnet?

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

That’s no so in the apple paradox. 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 statement means, these people could come to a different probability. Assuming no mistakes in calculation—a large assumption for complex statements—then each person would give correct answers to 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 of a sack. This time does not have to be constant; 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, conditional on these assumptions, 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!”

Sound familiar?

A version of this post first appeared on 30 October 2017.

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