If we let O = event object is present and N = event object not present we can set up the following
$ \textrm{Pr}[D_k | O] =\tbinom nk (p_2)^k(1-p_2)^k $
$ \textrm{Pr}[D_k | N] =\tbinom nk (p_1)^k(1-p_1)^k $
and what we're looking for is to find the value of k that means that it is more likely that an ojbect is there rather than not there. In other words:
$ \frac{\textrm{Pr}[D_k | O]} {\textrm{Pr}[D_k | N]} > 1 $
If you set up the equations appropriately, you can take the log of both sides to find a value of k. It gets pretty messy, but I got the following:
$ k > \frac{-n \cdot ln(\frac{1-p_2}{1-p_1})}{ln(\frac{p_2(1-p_1)}{p_1(1-p_2)})} $
