What is the probability that a positive integer not exceeding 100
selected randomly is divisible by 3?

Probability (P)of an event (E) is given by the formula :

        Number of Outcomes favourable to E     N(E)
P(E) = ------------------------------------ = -----
        Total number of possible outcomes      N(T)

provided that chances of every event are equal.

So, from the data you have provided :

N(E)= Total number of positive integers from 1 to 100 that are
divisible by 3
We know that 3 X 33 = 99 . So there 33 integers not exceeding 100 that
are divisible by 3
Therefore, N(E) = 33

N(T) = Total count of positive integers not exceeding 100
 We know that 0 is neither positive nor negative. So the first
positive integer is 1.And the last integer not exceeding 100 is 100.
Therefore, N(T) = 100

So, We have :
  P(E) = N(E)/N(T) = 33/100 = 0.33

Hope this helped.
If you need any clarifications, just ask!
:)

Yep, 1/3 it is

The answer to 
What is the probability that a positive integer selected randomly is
divisible by N?

is of course 1/N where N is a positive integer

asitya2k, it is NOT 1/3, rather the correct answer is 33/100.

33/100 is NOT 1/3.

The answer would only be 1/3 if the number 100 was excluded from the
sample space. However, it is not.

Also, yaborboichev, your equation is flawed and is not applicable in
this situation. Your rule (1/N) only works when the upper end of the
sample space is actually divisible by N. In this case it is not, so
your rule gives the wrong answer, 1/3, instead of the correct 33/100.