Suppose I have a continuum of agents $i \in [0,1] $ where each agent i takes the action $x_i$ where
$x_i = 1$ if $ \epsilon_i >-a$ and 0 otherwise
Assume that $ \epsilon_i $ has a standard normal distribution, $N(0,1)$
I want the probability that at least half the agents chose $ x_i =1$
What I have:
Let the prob that one agent chooses $ x_i =1$ be $p$, this is easily found.
We know that as n goes to infinity, the distribution of the choices of the agents goes to a normal distribution by the CLT with mean p and variance p(1-p)
Thus,the answer should be $P(X>0.5) = 1 - \Phi(\frac{0.5-p}{\sqrt{(p(1-p)}})$
Where $\Phi$ is the standard normal CDf
Thoughts? It doesn't took right.