Joint Default Probability

The Problem

Two bonds default independently with probabilities $p_1$ and $p_2$ over a 1-year horizon.

1. What is the probability that both bonds default? At least one defaults? Exactly one defaults?
2. Now suppose defaults are not independent — they're coupled by a Gaussian copula with correlation $\rho$. Specifically, let $X_i={\Phi }^{-1}(U_i)$ where $U_i\sim \text{Uniform}(0,1)$ and $(X_1,X_2)$ is bivariate normal with correlation $\rho$. Bond $i$ defaults iff $X_i<{\Phi }^{-1}(p_i)$ (i.e., $U_i<p_i$). Compute the joint default probability as a function of $\rho$.
3. What happens as $\rho \to 1$? As $\rho \to -1$? Why is this model infamous (hint: 2008)?