P-value based on the bin presenting the highest excess

The naïve solution to this paradox is to calculate a p-value using only the bin presenting the highest fluctuation. This approach would give a very small number ( $0.5\times 10^{-36}$) in our 1000 bin example, and would remain below the $1\%$ threshold even if the spike has only 5 events over a background of 1. Applying this reasoning to the ROG data we get

$$\left.\mbox{p-value}\right\vert _{\mbox{max}} = P(n_c \ge 4\,\vert\,{\cal P}_{\lambda_B=0.57}) =2.8 \times 10^{-3} \,,$$ (2)

a p-value which may be considered `significative'. (Note that, if we used the observed background of Fig. 1, that we do not believe it is the correct number to use, the p-value would be $5.2 \times 10^{-4}$ .)