What correlation does to voting

Model votes as $V_i = \mathbb{1}[Z_i = 1]$ where, with probability $\rho$, all votes equal a single shared draw, and with probability $1-\rho$ they are independent.

1. Compute the probability the majority is correct as a function of $\rho$, $p$, $k$.
2. Take $k\to\infty$ and interpret the limit.
3. Define an effective number of independent votes and compute it for $\rho = 0.5$, $k = 100$.