In Emerald, I found my first shiny, a Poochyena. I wasn't expecting it as I was running west of Lilycove to visit the Safari Zone. Also, I was scared of it using Roar, so I just used my Master Ball.
Btw, Apostrophe, the shiny ratio is determined by this formula:
(Trainer ID) xor (Secret ID) = E
(First byteword of personality value) xor (Second byteword of personality value) = F
If E xor F is less than eight, then the Pokémon is shiny.
So, 8/65536 chances reduces to our lovely 1/8192 chance. It was never 1/65536.