The problem asks us to find the probability that a positive integer between 1 and 100 (inclusive) is not divisible by 2, 3, or 5.

To solve this problem, we need to find the number of positive integers in the given range that are not divisible by 2, 3, or 5, and then divide this number by the total number of positive integers in the given range (which is 100).

Step 1: Finding the number of positive integers not divisible by 2, 3, or 5
To find the number of positive integers not divisible by 2, 3, or 5, we need to subtract the number of positive integers divisible by 2, 3, or 5 from the total number of positive integers in the given range.

Divisible by 2:
To find the number of positive integers divisible by 2, we can divide the total number of positive integers in the given range (100) by 2, which gives us 50. However, we need to exclude the numbers divisible by both 2 and 3 (i.e., multiples of 6). There are 16 such numbers (6, 12, 18, ..., 96), so we subtract 16 from 50 to get 34.

Divisible by 3:
To find the number of positive integers divisible by 3, we can divide the total number of positive integers in the given range (100) by 3, which gives us 33. However, we need to exclude the numbers divisible by both 2 and 3 (i.e., multiples of 6). Since we have already excluded these numbers in the previous step, we don't need to subtract anything.

Divisible by 5:
To find the number of positive integers divisible by 5, we can divide the total number of positive integers in the given range (100) by 5, which gives us 20. However, we need to exclude the numbers divisible by both 2 and 5 (i.e., multiples of 10) and the numbers divisible by both 3 and 5 (i.e., multiples of 15). There are 9 such numbers (10, 20, 30, ..., 90), so we subtract 9 from 20 to get 11.

To find the number of positive integers not divisible by 2, 3, or 5, we subtract the numbers of positive integers divisible by 2, 3, or 5 from the total number of positive integers in the given range: 100 - 34 - 33 - 11 = 22.

Step 2: Finding the probability
To find the probability, we divide the number of positive integers not divisible by 2, 3, or 5 by the total number of positive integers in the given range (100): 22/100 = 0.22.

Therefore, the probability that a given positive integer lying between 1 and 100 (inclusive) is not divisible by 2, 3, or 5 is 0.22.