Deliberate mistake: Elliot threatens Olivia about her testing positive for Oxycontin because she drank a peppermint mocha he brought her. If she tests positive she loses custody of her child. The thing with Oxy is that it's in one's system for a maximum of 4 days. Could be even less if only a small amount was taken and somebody exercises and drinks lots of water. Somebody could also eat poppyseeds and that would give a false positive. She could say she ate a poppyseed bagel. (00:14:00 - 00:16:00)
Episode #1.3 - S1-E3
Deliberate mistake: When the judge shoots himself, the revolver slides across the table, to fake his murder by someone else. But there is no mechanism in place that explains how he does this, making it appear as if he is able to accurately throw/slide the gun after shooting himself. In the original novel, he used a rubber band to achieve the effect, but he is not shown setting it up here.