Suppose that parameter $theta$ characterizes the probability that an individual takes an action. In the population, $thetasim U(0,1)$. Individual i has probability $pi(theta_i)$ of taking the action. Suppose further that

begin{equation}

pi(theta_i)=begin{cases} pi_1 & if & thetaleq c\

pi_2 & if & else

end{cases}

end{equation}

What is the correct formula for the average probability? Is it

begin{eqnarray}

pi&=&int pi_i di\

&=&int_{0}^c pi_1 di +int_{c}^1 pi_2 di

end{eqnarray}

or is it

begin{eqnarray}

pi&=&int pi_i di\

&=&Prob(thetaleq c)int_{0}^1 pi_1 di +Prob(thetageq c)int_{c}^1 pi_2 di

end{eqnarray}