= Solution
Assume independent arrivals at constant rate $r=2000\,\mathrm{s^{-1}}$, so the count follows a <Poisson distribution>. In $1\,\mathrm{ms}$ its mean is $\mu=2$. Therefore
$$
\boxed{P(N<3)=e^{-2}\left(1+2+\frac{2^2}{2}\right)=5e^{-2}\simeq0.6767.}
$$
Over observation time $T$, the rate estimate is $\widehat r=N/T$ and its standard deviation is $\sqrt{r/T}$. Thus the fractional one-standard-error uncertainty is $1/\sqrt{rT}$. To make this strictly less than ten per cent requires
$$
\boxed{rT>100,\qquad T>0.050\,\mathrm s.}
$$
The mean current magnitude is $er\simeq3.20\times10^{-16}\,\mathrm A$. The accuracy statement needs a confidence convention: the boxed time uses one standard error, not a guaranteed bound on every realization. A Gaussian 95-percent relative interval would instead require approximately $(1.96/0.1)^2$ events, or $T\gtrsim0.192\,\mathrm s$. Dead time and correlated arrivals would need another count model; the mean rate alone does not establish Poisson statistics.
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