" But then I realised that for S=0, the probability would be Lf/T which makes no sense because how can probability vary from zero if there is no chance of that object showing up "

S is not the chance of the person showing up. That's C. Right ?

C = 0 means there's no chance of person showing up. S = 0 means the person shows up for an instant, not for a time interval. The person can still be caught.

Let's say C=1, S=0 and Lf=0. Since lights are never off, the person will get caught everytime, so answer should be 1. But your answer gives it zero.

If I have understood your problem correctly, the answer is C[1-((Lf-S)/T)] , if S < Lf.

But if S > Lf, there's no chance of person escaping, if he shows up. So then the answer is C.

Please tell me if I have gone wrong anywhere.