Computing conditional probability with uniform distribution


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}