Consider the $majority$ function on three bits, $\textbf{maj}: \{0, 1\}^3 \rightarrow \{0, 1\}$ where $\textbf{maj}(x_1, x_2, x_3)=1$ if and only if $x_1+x_2+x_3 \geq 2$. Let $p(\alpha)$ be the probability that the output is 1 when each input is set to 1 independently with probability $\alpha$. What is $p'(\alpha) = \frac{d}{d\alpha} p (\alpha)$?
- $3 \alpha$
- $\alpha^2$
- $6\alpha(1-\alpha)$
- $3\alpha^2 (1-\alpha)$
- $6\alpha(1-\alpha)+\alpha^2$