Answer
${\bf 0.135}$
Work Step by Step
The problem deals with finding the probability ratio of a particle in the classically forbidden region beyond a boundary at $ x = L $. The wave function decreases exponentially in this region, and we need to compute the ratio of probabilities at two different positions: $ x = L $ and $ x = L + \eta $, where $ \eta $ is a small distance.
The probability of finding a particle in a small interval $ \delta x $ around a position $ x $ is given by
$$
\text {Prob(in } \delta x \text { at position } x\text {)} = |\psi(x)|^2 \delta x
$$
where $ |\psi(x)|^2 $ is the square of the wave function at $ x $, representing the probability density.
The ratio of the probability at $ x = L + \eta $ to the probability at $ x = L $ is:
$$
\frac{\text {Prob(in } \delta x \text { at } x = L + \eta)}{\text {Prob(in } \delta x \text { at } x = L)} = \frac{|\psi(L + \eta)|^2 \delta x}{|\psi(L)|^2 \delta x}
$$
$$
\frac{\text {Prob(in } \delta x \text { at } x = L + \eta)}{\text {Prob(in } \delta x \text { at } x = L)} = \frac{|\psi(L + \eta)|^2}{|\psi(L)|^2}\tag 1
$$
For $ x \geq L $, the wave function $ \psi(x) $ decreases exponentially which is given by
$$
\psi(x) = \psi_{\text {edge}}\;\;e^{-(x - L)/\eta}
$$
where $ \psi_{\text {edge}} $ is the wave function at the boundary $ x = L $, and $ \eta $ is the decay length.
So, At $ x = L $, the wave function is
$$
\psi(L) = \psi_{\text {edge}}\tag 2
$$
and at $ x = L + \eta $, the wave function is
$$
\psi(L + \eta) = \psi_{\text {edge}} e^{-1}\tag 3
$$
Plug (2) and (3) into (1);
$$
\frac{\text {Prob(in } \delta x \text { at } x = L + \eta)}{\text {Prob(in } \delta x \text { at } x = L)} = \frac{|\psi_{\text {edge}} \;\;e^{-1}|^2}{|\psi_{\text {edge}}|^2} = e^{-2}
$$
The value of $ e^{-2} $ is approximately:
$$
\frac{\text {Prob(in } \delta x \text { at } x = L + \eta)}{\text {Prob(in } \delta x \text { at } x = L)} =\color{red}{\bf 0.135}
$$
