$\rho = \psi_1^* \psi_2$ is the probability density for transition from state 2 to 1.  I am having trouble interpreting the probability though $$dP_{1\rightarrow 2}=\psi_1^* \psi_2 d^3 r $$ does this mean the probability in a region of volume $d^3 r$ to go from state 2 to state 1? That does not make sense at all!

**EDIT:** I agree with Marek's answer. But why is this not the probability density. If we have two states:  
$$ -i \hbar \partial_t \psi_{1,2}= -\frac{\hbar^2}{2m} \nabla^2 \psi_{1,2} + V\psi_{1,2}$$

After making standard manipulations (taking conjugate, multiplying by conjugate, subtracting) we would get
$$i\hbar \partial_t ( \psi_1^*\psi_2) +\frac{\hbar}{2mi} \nabla\cdot \left( \psi_1^* \nabla \psi_2 - (\nabla \psi_1^* )\psi_2\right)=0$$
This seems consistent with the continuity equation: $$\partial_t \rho +\nabla.\mathbb{j} =0$$ However, we seem to be changing its relation to the probability by making the probability $P=\int \rho^*(r,r') \rho(r,r') d^3r d^3r' $ instead of the regular $P=\int \rho(r) d^3r$