California does not offer true expungement, only post-conviction dismissal under Penal Code sec 1203.4. That remedy does not delete, remove or seal your record, but it adds an additional entry re the fact of the dismissal. You are not eligible for sealing of the record if you were an adult at the time of your conviction. The fact that you were severely impaired by alcohol is irrelevant for these purposes.
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The age of the conviction is on your side, 2004, and I am assuing it was for a misdemeanor charge. You certainly appear to be eligible for the relief you are seeking. As to how likely it is that the Judge will grant it, that can get complicated. Factors include your performance on probation and whether you were rearrested and convicted of new offenses. I have two offices close to you in Pasadena, and my Downtown office is probaly 11 minutes away if you would like a consultation. You can contact me through my website:
Short answer: Yes!
Longish answer: Yesssssss!! But seriously, yes. If you haven't been in trouble since, and followed all of the court's orders regarding your conviction, you will likely be able to get this charge expunged.