How many rounds of blackjack do you have to simulate to get accurate outcome probabilities? Say you play blackjack a lot and record how often each outcome occurs (win, push, lose, win $2x$, lose $2x$, $+3x$, $-3x$, $+4x$, $-4x$, $+1.5x$) (keep in mind you can double and split).
How many rounds would you have to play to be $90\%$ certain that the probabilities you record are within $0.001\%$ of the actual probabilities?
edit
Assuming you play using the basic strategy which is consistent from round to round (no card counting).
 A: Your 90% confidence is close enough to 2SD (that actually matches better with 95% confidence, but let's forget that). For the purposes of estimating the probability of a single outcome a single round is a Bernoulli trial. Let's say that the true probability of outcome A is $p$. Let $X$ be the random variable that counts how many times outcome $A$ occurred in $N$ rounds of simulation. The expected value of $X$ is then $E(X)=Np$. The variance of $X$ is $\sigma^2=Np(1-p)$. Therefore $\pm$2SD-interval of $X$ has halfwidth
$
2\sigma=2\sqrt{Np(1-p)}.
$
When we estimate $p$ with $X/N$, the 2SD error would then be 
$$\Delta p=\frac{2\sigma}{N}=\frac{2\sqrt{p(1-p)}}{\sqrt{N}}.$$
You asked for $\Delta p<10^{-5}$, so you want
$$
\sqrt{N}>2\cdot10^5\sqrt{p(1-p)}\Longleftrightarrow N>4\cdot10^{10}p(1-p).
$$
After you have done enough many rounds of your simulation, you will have a fairly good ideas of the value of $p(1-p)$, so you can use the above formula.
This formula probably has some inaccuracies, but for large $N$ they are hopefully negligible.
You see that getting such an accurate estimate for a probability by Bernoulli trials takes a large number of them. When I was doing channel coding simulations in a previous job we used a ball park figure of requiring $X>200$ before stopping a simulation. This could be trusted to give one significant figure for $p$. We were mostly interested in $\log_{10}p$ with a margin of error something like $\pm 0.1$, so that was about right :-)
