I'm afraid that's a very broad question! It depends on what procedure you are having done to the shoulder arthroscopically. It also depends what you mean by success - relief of pain, increased strength, stability, return to sport?
In very broad terms, no procedure carries an absolute guarantee of success. Also, in principle, the success rate of a procedure done arthroscopically compared to being done open is generally equivalent.
Very roughly, I give patients an 8 to 9 out of 10 chance of whatever shoulder arthroscopic procedure they are having working. However, your surgeon will perhaps be able to give you more specific and personal figures based on exactly what procedure you are having.
I hope this helps.