Mosquito Bites

The IIT Screening Exam yesterday, I must say was tougher than the 2003 and 2004 versions; mine was far from satisfactory. One exam over, and that means one less to worry about.
Anywayz it was sad to see not even one correct answer for the questions I posted on the blog. I've got another question with me, but i've no hope. I dont think I will get a correct answer to this one too.
here is the question anywayz: -

There are 'N' concentric cubical boxes, (a box inside a box, like that).
Each of these boxes has a small door which can open or close. Imagine a mosquito to be trapped inside the smallest box. Initially, all the doors are closed (i.e. at t=0). At t= 1s, 3 doors open randomly. the next second, the doors shut, and 3 doors (same or different) open. Assuming the mosquito is intelligent enough that it moves from a smaller room to a larger room only, when the doors are open, and the probability that a door opens at any instant is 'P', find the probability that the mosquito comes out of the 'N' boxes after 'K' seconds. (assume time taken by the mosquito to move from one box to another is negligible)