5.4 Consider a decaying radioactive source observed in a time interval of duration T D 15 s; N is the number of total counts, and B is the number of background counts

(assumed to be measured independently of the total counts):

(N D 19 counts B D 14 counts

:

The goal is to determine the probability of detection of source counts S D N B in the time interval T.

(a) Calculate this probability directly via:

Prob(detection)D Prob(S > 0/data)

in which S is treated as a random variable, with Gaussian distribution of mean and variance calculated according to the error propagation formulas. Justify why the Gaussian approximation may be appropriate for the variable S.

(b) Use the same method as in (a), but assuming that the background B is known without error (e.g., as if it was observed for such along time interval that its error becomes negligible).

(c) Assume that the background is a variable with mean of 14 counts in a 15 s interval, and that it can be observed for an interval of time T  15 s. Find what interval of time T makes the error B15 of the background over a time interval of 15-s have a value B15=B15 D 0:01, e.g., negligible.