= Solution
<Ohm's law> gives the mean <electric current>
$$
\boxed{I=\frac VR=10^{-12}\,\mathrm A=1\,\mathrm{pA}.}
$$
For an estimate, assume the <shot noise counting model>: electron transfers are independent, so the number recorded in an interval $\tau=1\,\mathrm s$ has a <Poisson distribution> with <expectation value> $\mu=I\tau/e=6.2415\times10^6$ and <variance> $\mu$. One measurement has relative <standard deviation> $1/\sqrt\mu=4.00\times10^{-4}$. Averaging $M$ independent intervals reduces this to $1/\sqrt{M\mu}$, so
$$
M\geq\frac{1}{\mu(10^{-5})^2}=1602.18,\qquad \boxed{M=1603\text{ one-second measurements},}
$$
about $27$ minutes in total.
This is a one-<standard deviation>, <shot noise> estimate. A resistance and mean <voltage> alone do not specify the actual current fluctuations: a real <resistor> also has <Johnson-Nyquist noise>, and electron-transfer correlations can change the <shot noise>. Its <temperature>, measurement <bandwidth> and <noise> model would be needed for an unconditional acquisition time.
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