I am stuck in this problem for some time now :( :(. it will be very helpful if anyone can give me some hint.

problem link : problem

My idea :

lets consider each light individually.

Let p is the probability of selecting the light. this p can be found easily by calculating each dimension of the 3d cube separately.

Now the expectation for a light will be sum over i(1 to k) : p^i * (1-p)^(k-i). (K is the number of turns) Now we only need to add the terms where i is odd as only then the light will be ON.

calculating this for all the bulbs is obviously time out :( :( :( is there any close form? i can't find it :( :( :(