If charges are dropped, it's not a conviction. However, before completing it, you should get a copy of the court records and see what it has. The judge may have dropped the public intoxication to a lesser charge, but you still could have been convicted of something.
Better safe than sorry.
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No. A dropped or dismissed charge is not considered a conviction. You may want to double check and look up your charge to see what the disposition was prior to completing your job application.
When the charge is dropped/dismissed, that is not a conviction. It is quite common in San Luis Obispo County to do 20 AA meetings in exchange for dismissal of a Public Intoxication charge. You should have received a piece of paper showing the outcome of your case. If you still have that, a lawyer could look at it and confirm for you that you actually did get your charge dismissed.
It's not a conviction if dropped. Double check and get the minute order to confirm.
Since it's not a conviction, you can mark "no" you have never been convicted. However, if you were arrested, you would still have to answer "yes" if it asks if you were arrested.
If you did not plead guilty or no contest to the charge, then it was not a conviction. This sounds like you were given the option of a dismissal in exchange for some conditions. If you are leaving out a plea of guilt was made, then this changes the equation.