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Are there any seemingly simple probability question that are actually intractable?

Are there any good examples of a seemingly simple probability problem, which is actually intractable?

I am trying to motivate the use of simulation, and would like to come with an example of when it is necessary that is accessible. The hope being something like:

"Intuitively it seems easy enough to model the amount of aces left after a round of poker, but due to $x$, $y$ and $z$, this is actually infeasible to calculate analytically".

But I am struggling to find a good / simple example.

Any help would be appreciated.