= Solution
Assume independent background events at a known constant rate, so the background count follows a <Poisson distribution>. In five months its mean is $\mu_b=0.3\times5=1.5$. The <null hypothesis> of background alone assigns the observed upward fluctuation the <P-value>
$$
p=P(N\geq8\mid\mu_b=1.5)
=1-e^{-1.5}\sum_{n=0}^{7}\frac{1.5^n}{n!}
\simeq1.696\times10^{-4}.
$$
Thus
$$
\boxed{p\simeq1.7\times10^{-4}>10^{-6}:\quad\text{the discovery criterion is not met}.}
$$
The tail includes counts above eight because they would be at least as discrepant with the background-only prediction. Even the probability of exactly eight, $e^{-1.5}1.5^8/8!\simeq1.42\times10^{-4}$, exceeds the specified threshold, though it is not the appropriate tail test. The <Poisson tail discovery probability> is a probability of data under a background model; it is not a posterior probability that the background model is true. This interprets the question's discovery rule in its usual <statistical significance> sense.
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