Exterior probability of a symmetric finite-well state ($P_{\mathrm{out}}=C^2{e}^{-qa}/q$)

If a normalized wavefunction has exterior tails $C{e}^{-q|x|}$ outside $|x|<a/2$, integration over both tails gives the title probability. For an even cosine interior $A\cos (kx)$, the remaining normalization contribution is $A^2[a/2+\sin (ka)/(2k)]$. Tail amplitudes defined at the edge rather than at the origin absorb the factor $e^{-qa/2}$, so their probability formula looks different despite describing the same state.