No evens with no perfect cube!!!

First given condition is that number is not a cube.. which leaves us with 990 (=1000-10) possibilities [1³,2³,...10³=1000]

Now, only square numbers have odd divisors(NOT EVEN divisors)

From 1-1000, 31 sqaures are there (1², 2², ....31²=961)

But out of these 1²=1=1³ ; 8²=64=4³ and 27²=729=9³ are omitted from first condition which leaves us with 28 squares.

So required probability is 28/990=0.028

The only numbers with an odd number of divisors are perfect squares. There are 31 squares between 1 and 1000, inclusive. The squares that are also cubes in the set of squares between 1 and 1000, inclusive, are 1, 64, and 729. Thus, the probability is $\frac{31-3}{1000} = \boxed{0.028}$