Answer
$ {\bf 0.0095}\%$
Work Step by Step
We know that the probability that an electron will tunnel through a
0.45 nm gap from a metal to an STM is given by
$$P_{\text{tunnel}} = e^{-2w/\eta} \tag 1$$
So we need to find $\eta$ since we know $w=0.45$ nm.
We know that
$$\eta = \frac{\hbar}{\sqrt{2m(U_0 - E)}} $$
where the $U_0-E=E_0$ where $E_0$ is the work function energy that must be supplied to lift an electron out of the metal.
$$\eta = \frac{\hbar}{\sqrt{2mE_0}} $$
$$\eta= \frac{(1.05 \times 10^{-34} )}{\sqrt{2(9.11 \times 10^{-31} )(4.0 \times 1.60 \times 10^{-19} )}} = \bf 9.72 \times 10^{-11} \, \text{m}
$$
Plug into (1);
$$P_{\text{tunnel}}= e^\frac{-2(0.45\times 10^{-9} ) }{ (9.72 \times 10^{-11} )} $$
$$P_{\text{tunnel}}= 9.5 \times 10^{-5} = \color{red}{\bf 0.0095}\% $$
