In response to your comment, note that if we set $B''=0$ and include the time evolution factor $e^{-iEt/ \hbar}$, then your equations model the situation you want since $A$ is the amplitude of a wave travelling to the right in region 1 (towards the barrier) and there is no wave travelling to the left in region 3. The amplitude $A''$ can be seen as telling us how much of the incoming wave has been transmitted, so it is interesting to quantify $A''$ in terms of $A$. 

To see that $T = {|A''|^2}/{|A|^2}$ can be seen as the probability that the incoming wave has been transmitted, note that the "probability" to find the particle in any interval of length $\ell$ in region 3 is $|A''|^2 \cdot\ell$ (carry out the very simple integral), whereas if there was no potential barrier (i.e. $V=0$) then the probability to find the incoming matter wave in the same interval would be $|A|^2 \cdot\ell$.
So the ratio $|A'' |^2/|A|^2 =|A'' |^2 \cdot\ell/(|A|^2 \cdot\ell)$ can be seen as the transmission probability.

Note that the argument above is not literally correct, which is why I put quotation marks around "probability": the solutions to this problem cannot be normalized, and so it doesn't make sense to talk about probabilities of finding states in some interval. But this is a different matter - hopefully this argument still helps a little bit.