The detection limit is the answer to the question - what is the
minimum number of counts we can be confident of detecting in a given
time?

Suppose measure a weakly active sample, obtaining a count C. A
measurement of background gives B. So the count due to the sample
itself, S = C - B.

At the detection limit the average value of S: <S> = L_{D}.

The plot shows the distribution P(S) of repeated measurements. They are
distributed about L_{D} with a standard deviation s_{D}.

If the critical limit is L_{C,} the detection limit L_{D} =
L_{c} + ks_{D }= k^{2} + 2L_{C.}

L_{D} depends on background
B and your choice of k.

The fractional area of P(S) above L_{C} gives the probability that any
measurement of this source will be significantly above background,
i.e. not consistent with zero.

The area gives us our confidence level, which depends on our
choice of k.