Quote by TaliskerBA
Since there are k particles emitted and we want to know the probability that r of them have been detected then using binomial distribution:
p((Y=r) ∩ (X=k)) = [k choose r](p^r)(q^(kr))
I know this is wrong but I can't quite work out how to tie it all together...

You wrote down a formula assuming that k particles have been emitted. Therefore this is not the joint probability, but it is still relevant to the conditional probability that you want to compute...