Geometric Mean of Random Gains via LLN and CLT

Let $X_1, X_2, \ldots$ be i.i.d.\ with $P(X_i = 1) = \tfrac{1}{2}$ and $P(X_i = 0) = \tfrac{1}{2}$. Define the geometric-mean-like quantity $$Y_n = \left(\prod_{i=1}^{n} (1 + X_i)\right)^{1/n}.$$

(a) Find $\lim_{n \to \infty} Y_n$ almost surely.

(b) For $n = 200$, use the CLT to approximate $P(Y_{200} > 1.45)$.

You may use: $\ln 2 \approx 0.6931$, $\Phi(1.02) \approx 0.8461$.