What is the chance of rolling TWO 6-sided dices and getting one even and getting one odd.

I think it's:

1/2 x 1/2 = 1/4

BUT couldn't it also be:

1/2 x 1/2 = 1/4 x 2 = 1/2 ???

Think like this: You roll the dices one by one. The first dice will be even or odd. Regardless of the first one, the second one will come the opposite of first by the chance of 1/2.

Or think like this: Again you roll them one by one. If first is even(1/2), the other must come odd(1/2). Total chance of 1/2 * 1/2 = 1/4. If the first come odd(1/2), the other has to come even(1/2). Second option: 1/2 * 1/2 = 1/4. These two cannot happen at once, so we can add them-1/4 + 1/4 =1/2.

I think you thought that "one even is 1/2, one odd is 1/2 and a total chance of 1/4." But the sequence of them can change. Odd-even or even-odd.

The first way of calculating is a very practical way, and can be used in a lot of probability questions:
Waiting for the first one to happen, and then calculate the other one.

And, there is a good way, too. If you do not have many dices, or many coins to flip, you can just write the whole events, like in here:

First: even odd even odd

Second: even even odd odd

There are 4 events, two desired, so it is 2/4=1/2.

1/2

4 choices, 2 evens two odds and one even and one odd twice