Probability of 8 in 65536; simplified as 1/8192 (2-13, or approximately 0.012207%), just as in Generation II (gold/silver/crystal). The same as all the rest of the shinies.
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if you want to evolve budew you will need to make it happy and it has to be daytime. if you then wnt to evolve your Roselia into a Roserade, you have to give it a shiny stone
its not rare just shiny
Shiny Zubat are a rare instance of Zubat that are colored different. There is a very rare chance that any Pokémon you encounter in the wild will be shiny.
Budew does not evolve from leveling it up, Budew evolves from friendship. you will know it has full friendship with poketch app friendship checker, which you get from the president of the poketch company, and when budew has two giant hearts, it has full friendship. I suggest giving it a soothe bell, feeding it pofffins, walking with it in the area north of hearthome, and battling with it. If you want a roserade use a shiny stone which you can find in the side entrance of iron island NOTICE: Budew will only evolve from friendship DURING THE DAY
it is the rarest shiny ever in history only one even close to that is shiny arceus