The probability that a given positive integer lying between 1 and 100 (both inclusive) is NOT divisible by 2, 3 or 5 is _________.

The total number of positive integers from $1$ to $100$ is $N = 100$.

Let $A_2, A_3, A_5$ be the sets of integers divisible by $2, 3, 5$ respectively.

Using the Principle of Inclusion-Exclusion:

• $|A_2| = \lfloor 100/2 \rfloor = 50$

• $|A_3| = \lfloor 100/3 \rfloor = 33$

• $|A_5| = \lfloor 100/5 \rfloor = 20$

Pairwise intersections:

• $|A_2 \cap A_3| = \lfloor 100/6 \rfloor = 16$

• $|A_2 \cap A_5| = \lfloor 100/10 \rfloor = 10$

• $|A_3 \cap A_5| = \lfloor 100/15 \rfloor = 6$

Triple intersection:

• $|A_2 \cap A_3 \cap A_5| = \lfloor 100/30 \rfloor = 3$

The total number of integers divisible by at least one of $2, 3,$ or $5$ is:

Therefore, the number of integers not divisible by $2, 3,$ or $5$ is:

The required probability is: