 Solution

Assume independent background events at a known constant rate, so the background count follows a Poisson distribution. In five months its mean is ${\mu }_b=0.3\times 5=1.5$. The null hypothesis of background alone assigns the observed upward fluctuation the P-valueThusThe tail includes counts above eight because they would be at least as discrepant with the background-only prediction. Even the probability of exactly eight, $e^{-1.5}1.5^8/8!\simeq 1.42\times 10^{-4}$, 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.