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.
 A: The survival function $S_{t}$ is a quantity of interest in many (most?) kinds of event history analysis. It is commonly estimated, and 'survival curves' depicting $S_{t}$ versus time are often used to compare the cumulative probability of events among different groups. Statistical comparisons are often facilitated by inference—things like hypothesis test and confidence intervals.
I and a few statisticians have struggled with several different approaches to provide an asymptotic analytic estimator of the sample variance of the survival function ($\sigma^{2}_{\hat{S}_{t}}$) in discrete time event history models (a la logit hazard, probit hazard, etc. models), which would be useful to construct hypothesis tests and confidence intervals.
It turns out that—as best I understand it—that while it is possible and common to estimate the asymptotic variance of sums of random variables (like the sample mean), the asymptotic variance of products of random variables is a tricky sticky wicket to estimate.
$$\hat{S}_{t} = \prod^{t}_{i=1}{1-\hat{h}_{i}}$$
where $\hat{h}_{t}$ is the discrete time hazard function at time $t$.
We have more or less given up on an asymptotic estimator of the variance of that puppy, and declared that numerical techniques like bootstrapping seem to be our best bets.
A: You've got 5 variables and you're doing a "multivariate" analysis. You assume multivariate normality and enjoy a complete data set. Then the maximum likelihood estimates of the mean and covariance matrix are closed form and easy to calculate.
Oh wait, you didn't want to assume joint normality. You meant to assume that, marginally, each of your variables follows a beta distribution. No big deal. There must be a multivariate analogue of the beta distribution with an arbitrary correlation structure, right? Well, you might be able to construct something, but I'll call it "intractable" for my level of patience. Here is a reddit post from someone trying to figure out something similar without much luck.
A: A simple probability problem that is intractable could be the following for a horse race.
If the horse trainer has a win rate of 25% and the jockey a 10% win rate and the horse has a 40% win rate what is the un-normalised probabilty of success of the horse in today's race?
The trainer has trained the horse to have a 40% success rate but will the rate fall in future races towards 25%? Will the jockey have a better chance than 15%, and by how much, on a horse that wins 40% of the time and a trainer that wins 25% of the time?
