Estimate π by throwing random points at a unit square, with 1000 points and with 100 000 points…

Check. The theory says that the spread is $4\sqrt{p(1-p)/n}$ with $p = \pi/4 = 0.785$, which is $1.64/\sqrt{n}$: 0.052 for $n = 1000$ and 0.0052 for $n = 100\,000$. The simulation gives 0.0488 and 0.0049. The ratio is about 10, which is $\sqrt{100}$.