I assume starting point are uniformly randomly taken from [0,T] (they could be randomly taken with another distribution), and that the length is delta

Consider the start of the first interval to be s1, then any other interval si will overlap with it (call it event A) if s1-delta < si < s1+delta, or in other words will overlap with probability

P(A)=P(max(0,s1-delta) < si < min(T,s1+delta))

which should be easy to determine in case of uniformity. The answer to the first question then amounts to make a product of these probabilities (independent events), while for the second one you have to work out the combinatorics, but this should be feasible.

I assume starting point are uniformly randomly taken from [0,T] (they could be randomly taken with another distribution), and that the length is delta

Consider the start of the first interval to be s1, then any other interval si will overlap with it (call it event A) if s1-delta < si < s1+delta, or in other words will overlap with probability

P(A)=P(max(0,s1-delta) < si < min(T,s1+delta))

which should be easy to determine in case of uniformity. The answer to the first question then amounts to make a product of these probabilities (independent events), while for the second one you have to work out the combinatorics, but this should be feasible.

Dear Ammar,

the only way to answer those questions is through Monte Carlo simulation. I recommend that you use a simulation tool (e.g. Excel if you want) and randomly generate the m intervals and measure for each iteration when they overlap and when they don't.

I suggest this because initially it seems a simple problem but it is not because you can have two possible outcomes: that your first interval s1 is located leaving only one free space (toward its right or left) and it can be placed leaving empty spaced both at the right and left of itself. The probabilities of placing subsequent intervals without overlapping will change them for both possible outcomes. What is more, the probability will vary between all possible positions of the initial interval s1.

If that is the case, I am pretty sure that the distribution you are looking for is a Beta Binomial (a Binomial where the probabilities of success keep changing according to a Beta distribution). However, figuring out which will be the Beta distribution of this Beta Binomial will require simulations anyway. So, if you are going to implement simulations (since they will depend on the values of s1 width and T, just keep simulating it until you get all your answers.

I hope it helps.

@Sebastien Destercke thank you very much:)

@P. Ballesteros-Pérez thank you very much, I've solved the problem as suggested by @Sebastien Destercke. I'll share it later :).Thank you again. 

Inspired by the answers by Sebastien Destercke  and P. Ballesteros-Pérez I have obtained the following result for  T> 3d,  where  d  is the length of the intervals contained in  [0,T].  First, the probability of the overlaping S1 by a  single   Si,  for   i=2,3,...,M, under the condition that the left end of  S1  occupies position  x (which varies between 0  and T-d)  equals as follows:

P(x) = 2d/(t-d)    if     d

@ P. Ballesteros-Pérez.

In order to be more precise: If the following probabilities of a success are acceptable,

P(x) = 2d/(t-d) if d

@ Ammar Hawbani

The introduction requires more exact formulation. One of the possible corrections is the following

The intervals are randomly placed within [0,T], that is the start point of each interval is randomly selected within [0,T-d], where  d  is the common lenth of the intervals.

Otherwise the intervals would be not contained in  [0,T]. 

Regards, Joachim

Dear @Joachim Domsta:

actually, i meant that the starting point of each interval is randomly selected within [0, T].:)

thanks a lot for your contributions. 

@ Ammar

OK, the proposed model then will work correctly  with   T  repleaced by   T+d.

Thanks, for thanks:)

Joachim