We know , here Poisson distribution is followed .So
P(k) = e-G . (G)k / k! where G is the number of frames generated in one frame transmission time(or frame slot time)
Here to find G we have to find number of frames that are generated in 1 slot time which is given as 40 ms.
Given , in 1000 ms no of frames generated = 50
So , no of frames generated in 40 ms = (50 / 1000) * 40
= 2
Hence the value of G obtained = 2
Now we solve the given parts one by one.
Solution to part a) :
P(success at first attempt) = P(0)
= e-G
= e-2
Solution to part b) :
P(success after k collisions) = [ P(failure) ]k P(success)
= (1 - e-G)k e-G
= (1 - e-2)k e-2
Solution to part c) :
Expected Number of attempts = 1 . P(1) + 2 . P(2) ............... infinite terms
= Σ k . P(k) [ Where P(k) is the probability of success at kth attempt ]
= Σ k . e-G (1 - e-G)k-1
This is an infinite arithmetico geometric progression.So finding this sum , we get :
Number of attempts(expected or mean number) = eG
= e2
= 7.39
= 8 [As number of attempts should be an integer and hence we need to round it to higher integer in this case ]