P-value based on the overall number of events

The expected number of events due to the null hypothesis $H_0 =$   ``only background'' is 27.4 ( $= 0.57\times 48$). Having observed 34 events we get:

$$\left.\mbox{p-value}\right\vert _{\mbox{integral}} = P(n_c \ge 34\,\vert\,{\cal P}_{\lambda_B=27.4}) = 12\%\,,$$ (1)

a value that it is not considered `significative'. However, the obvious criticism to this procedure is that we have only used the integrated number of coincidences, losing completely the detailed information provided by the time distribution. The problem can be better understood in the limiting case of 1000 bins, an expected background of 1 event/bin, and an experimental result in which 999 bins have contents which `nicely' (Poisson) fluctuate around 1, and a single bin exhibiting a spike of 31 counts. The p-value would be of 16%, accepting the hypothesis that the data are explained well by background alone.