## Monday, November 1, 2021

### Shuffling infinitely many cards

Imagine there is an infinite stack of cards labeled with the natural numbers (so each card has a different number, and every natural number is the number of some card). In the year 2021 − n, you perfectly shuffled the bottom n cards in the stack.

Now you draw the bottom card from the deck. Whatever card you see, you are nearly certain that the next card will have a bigger number. Why? Well, let’s say that the card you drew has the number N on it. Next consider the next M cards in the deck for some number M much bigger than N. At most N − 1 of these have numbers smaller than N on them. Since these bottom M cards were perfectly shuffled during the year 2021 − (M + 1), the probability that the number you draw is bigger than N is at most (N − 1)/M. And since M can be made arbitrarily large, it follows that the probability that the number you draw is bigger than N is infinitesimal. And the same reasoning applies to the next card and so on. Thus, after each card you draw, you are nearly certain that the next card will have a bigger number.

And, yet, here’s something you can be pretty confident of: The bottom 100 cards are not in ascending order, since they got perfectly shuffled in 1921, and after that you’ve shuffled smaller subsets of the bottom 100 cards, which would not make the bottom 100 cards any less than perfectly shuffled. So you can be quite confident that your reasoning in the previous paragraph will fail. Indeed, intuitively, you expect it to fail about half the time. And yet you can’t rationally resist engaging in this reasoning!

The best explanation of what went wrong is, I think, causal finitism: you cannot have a causal process that has infinitely many causal antecedents.

IanS said...

Non-conglomerability strikes again!

An interesting mathematical point: By your theorem (“Conditional, Regular Hyperreal and Regular Qualitative Probabilities Invariant Under Symmetries”), if I’m thinking straight, there are regular conditional and hyperreal distributions on the finite permutations of the natural numbers. So, formally, there are non-standard probabilities that match the intuitions of the post.

jqb said...

"And yet you can’t rationally resist engaging in this reasoning!"

Sure I can.

Alexander R Pruss said...

Ian:

Yeah, that's an interesting mathematical point.

swaggerswaggmann said...
This comment has been removed by the author.
swaggerswaggmann said...

This is a feature of infinity. Don't worry if it seems illogical, as you already use the end of infinity to try to get a misunderstanding.
Due to N and M already being infinite ...
Crap in , crap out.