Probability of seeing / proving something Hypothetically my landlord say that he never see me airing my apartment,
assumed he says he went on a day between 08:00 and 20:00 randomly 6 times 
spying on me if one of my windows is open each session he is looking at my windows
for lets say around 1 minute assume there is a minimum delay between each session for around 20 minutes.
now i claim to randomly open my windows for around 15 minutes 2 times the day between
04:00 and 23:00 with a minimum delay of 3 hours.
So the question would be:
how many days would the landlord need to prove without doubt(80%) that i never opened my windows
and how many days would it take him to catch a open window if my claim is right.
 A: Let's try this. X is a geometric rv if it is the number of trials until we see a success. Let's say that each day is a trial, and let's say that him seeing you open a window is a success. So all we really have to determine is the probability that he sees a window open on a given day. 
There are just too many restrictions on the problem for practical purposes to do this analytically. But I wrote a quick simulation for the problem and ran it 25,000 times, I am finding that the probability that the time windows overlap is approximately 0.16. Now let's answer your questions.

Question 1
We have to think about what you're asking and how we can answer this with Statistics. Suppose you're telling the truth, then there is about an 82.5 percent chance that he will you see you with an open window within the first 10 days.
$P(X > 10) = 1 - F_X(10) = 1 - (1 - (.84)^{10}) = 0.84^{10} = 0.1749$
So if he goes more than 10 days without seeing you're window open, he can be about 82.5% sure that you're not telling him the truth.

Question 2
The Expected Value of a geometric distribution is 1/p. So on average, it would take him about 6.25 days before seeing you with a window open.
Although for this distribution, I prefer the Median, which is 3.97 here. This tells us that 50% of the time, he will see you with a window open before 4 days have passed.

