Sharonmo, good question. I was solving it the same way and got it wrong, but came to this forum to see if an explanation existed for your approach.
Restating it, here's how I went about it: (I did it on 2 separate accoasions 2 weeks apart, and got it wrong both times. So, I am determinded to nail it down.)
1. Ways in which you can get xy to be even = E * E, E * O or O * E.
First, looking at set x, 2 Even #s * any # from y = Even. That makes it 2*3 = 6 cases.
Then looking at set y, 1 even # * any # from x = Even. That makes it 1*4 = 4 cases.
Even * even from either set is already covered in the 2 equations above. (Am I wrong?)
That makes it a total of 10 cases, out of 12. Therefore the P() = 10/12 = 5/6.
Please let me know what I am doing wrong conceptually, so I can nail such problems in the future.
TIA.