Compute the Std. Deviation of Multiple Monte Carlo Estimation of $\pi$

Question

For a school programming assignment, I am trying to compute the value of $\pi$ via the classic Monte Carlo estimation of $\pi$. In the experiment, we throw a variable number of darts at a circle that is inside a square. These darts are thrown at random $(x,y)$ points within the square. Knowing the number of darts in the circle and square, we can estimate the value of $\pi$.

I am given that the number of darts in the circle may be thought of as $\mathrm{Binom}[n, \pi/4]$, where:

mean $= \pi n/4$ and std. deviation $= \sqrt{\pi n/4(1- \pi/4)}$.

I'm then asked to run the experiment with 100 darts 1000 times and compute the std. deviation in the 1000 estimates of $\pi$. This is where I get confused. Given the formula for std. deviation how would I use it for 1000 values of $\pi$. The formula seems to only be looking for one value of $\pi$. Could anyone show an example calculation?

Answer 1

Define a computation as:

throw 100 darts, compute the number of darts in the circle: this is the result of the computation.

You are asked to make 1000 computations, and to compute the standard deviation of the sample consisting of the 1000 results of the computations.