If the primes between 1 and 6 are: 1, 2, 3, 5 then the composites are 4 and 6. The chances of rolling a prime are 4/6=2/3 and of rolling a composite are 2/6=1/3. So the combined probability is 2/3*1/3=2/9.
If 1 is neither a prime nor a composite, the combined probability changes to 3/6*2/6=1/2*1/3=1/6.