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)
Deliberate mistake: Inside the home that has been infested with Evil, the state police confront a possessed girl who attacks them. After some futile gunfire, the possessed girl is still standing. The female officer goes for her backup gun, which for some reason is a small over-and-under Derringer (meaning she has two shots maximum). She ends up firing the Derringer five times in just a few seconds, with explosive damage and without reloading. Director Sam Raimi probably staged the 5 gunshots on purpose, knowing that a two-round Derringer was a ridiculously-limited piece of firepower. Raimi probably did this as a tongue-in-cheek tribute to Ash's bottomless double-barrelled shotgun from Evil Dead 2 and Army of Darkness.
Deliberate mistake: Lee's old crew of three are shooting various caliber handguns at Chloe and Lee's sister. Chloe took cover behind a kitchen island. Kitchen islands are made of thin wood, the only piping in them is plastic and some brass/copper. All of those handgun calibers would have penetrated the kitchen island. (00:45:45 - 00:47:00)
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.