Usually the jury would be discharged if convictions were accidentally disclosed when they wouldn't otherwise have been given in evidence. It's possible that the jury wouldn't be ,but only because the judge thought that the convictions were so slight and so long ago, or so irrelevant and insignificant, that the jury could be safely told to disregard them and treat the defendant as of good character and not hold the convictions against him.
In the normal trial, previous convictions are only admitted into evidence if they are relevant to some issue in the case being tried; they may yet be excluded , if the judge thinks them far more prejudicial than probative i.e tending to prove anything; or if the defendant is so unwise as to claim that he is of good character, in which case it's only right that the jury should be told that he is lying about that.
(One hazard concerning inadvertent indication of bad character, that counsel may not be prepared for, is the client's witnesses. I once saw the mother of the defendant giving an alibi for her son. His counsel, wishing to establish the relevant date, asked her "Do you remember the police coming to your house to ask about your son and a stabbing?" to which she thought for a moment and replied "Which stabbing was that?" The case proceeded, in the hope that nobody on the jury drew the unfortunate inference!)