generating keys by using only part of longer random number

If I have a random number generator that e.g. generates 512bit long
keys but I only need 160bits of them. Would the quality of the
randomness then be compromised if I always just uses the first 160bits
of the key?

My feeling is that it should not make it less random if you look at it
from a statistically perspective.

Example using digits for simplicity (3 digit random string S, 3 digit
key K needed):


getting K=123 happens for S=12300 to S=12399
This is 100 times in 100000 draws = once every 1000 draws.
if S was only 3 digits long it would get K=123 only once in 1000 draws
= statistically the same as above.

But is my statistically approach to this correct when working with