Some Comments:

A = odd number on first, B = even number on second, C = sum is odd.The assumption that A and B are independent is fair enough. C is pairwisely independent with A and B respectively but that is not immediate. Also note that they are not jointly/mutually independent.I think these three events are independent on each other.

and the dice is fair so you have a discrete uniform distribution. But the events A, B, C are not asking you the probability of obtaining a specific number. So why 1/6?Because there are numbers 1-6,