What is the mistake in judgement that Pedro and Claudio makeabout Hero?
In William Shakespeare's Much Ado About Nothing, calling women sexist names in public may be "nothing" for men, but it is a major problem for women. Like most men of the day, Claudio and Don Pedro choose to believe a male's opinion of a woman instead of a woman's. In short, male reputation was to be protected against women's, so as such, Claudio and Don Pedro choose a Bastard's opinion over a woman's.
Claudio and Pedro are ensnared by John the Bastard because they have been conditioned to distrust women. The society believed that women were either virgins or whores, and when Don John shows who they think is Hero necking with Borachio, they accept it as fact.
They mistakenly do not question Hero or Beatrice, her bedfellow. They mistakenly choose to expose Hero as a whore on her wedding day, in front of her father and the guest. They even convince her father of her guilt: that's how easy it was to denounce women and be done with them.
I suppose that the mistake in judgement that these two make about Hero is that they are unable to properly understand her character. They think she is not as good a person as she is. I guess you could say that they lack faith in her as a person.
The way this comes out is that they do not trust her. When Don John sets up the scene where Hero is supposedly showing that she has been having sex with some man, the two of them believe that it could be her. If they really understood her, they would realize that she was not capable of behaving in that way when she was in love with Claudio.