Estimating Pi by Monte Carlo and the Law of Large Numbers

To estimate $\pi$, you draw $n = 10{,}000$ points $(X_i, Y_i)$ independently and uniformly on the unit square $[0,1]^2$. Define $Z_i = \mathbf{1}(X_i^2 + Y_i^2 \le 1)$, and let $\hat{\pi} = 4\bar{Z}$ where $\bar{Z} = \frac{1}{n}\sum_{i=1}^n Z_i$.

(a) Explain why $E[\hat{\pi}] = \pi$ and why $\hat{\pi} \to \pi$ almost surely.

(b) Using the CLT, find an approximate $95\%$ confidence interval for $\pi$ given that the observed $\bar{Z} = 0.7854$.

You may use $\Phi(1.96) \approx 0.975$ and $\pi \approx 3.1416$.