On average, most rpg groups meet once a week. A lot can happen in a week and a lot can happen in a game, therefore it is advantageous of everyone involved to keep notes on what happened just in case of situations like this. This is doubly true for the DM. Players are sneaky little cheats, and proud of it, so you got to stay on top.
And this part of the episode is hard to get any good shots of Mal and Jayne due to the helmets they are wearing. Inara on the other hand is all over the place, facial expression wise, in her first scene.
DM: You have eight rounds, better make the best of it.
Mal: Alright, let’s get started. I use the corrosive glue around the lock.
DM: Okay, you want to roll dodge, the door was trapped.
DM: Yep, and with that suit on in this being zero gravity you get a… -5 to dodge.
Jayne: Hey, we searched for trapped at the last game, I found it and disabled it with a 17.
DM: No you didn’t.
Jayne: Yes we did, I made a note of it, see.
DM: Did you just write that?
DM: Did he just write that?
Mal: Did you make a specific note that John hadn’t disabled it?
DM: Yes! [shuffling papers] wait… frell.