(Mathematicians may find this post painfully obvious.)

I read an interesting puzzle on Stephen Landsburg's blog that generated a lot of disagreement. Stephen offered to bet anyone $15,000 that the average results of a computer simulation, run 1 million times, would be close to his solution's prediction of the expected value.

Landsburg's solution is in fact correct. But the problem involves a probabilistic infinite series, a kind used often on less wrong in a context where one is offered some utility every time one flips a coin and it comes up heads, but loses everything if it ever comes up tails. Landsburg didn't justify the claim that a simulation could indicate the true expected outcome of this particular problem. Can we find similar-looking problems for which simulations give the wrong answer?  Yes.