Dice probability question

You roll a blue and a red die. How many outcomes have a blue 3 or an even sum? The dice are fair dice.

I know that you have a 1 out of 6 chance for a blue 3 and you have half of the results will yield an even sum 18 since we have 36 possible outcomes. How do you determine how many will have both to subtract from the number of outcomes that have either of these????? I know it is 3 but I did it by exhaustion and if the question had many more outcomes I would not be able to figure it out. Thanks again all of you for your helpful and kind assistance.

Dice probability question

Thank you Captain Black for your detailed answer. I really appreciate it!