All you guys complaining about the lack of burn card
This was intentional, it is a TV show! Obviously the writers know how to play Hold'em---you did not catch something they missed. Probably 90% of the viewers wouldn't know about burn cards. If a burn card was included in the script, either they would have to insert some very awkward expositional about burn cards, or Vince's "proof" would confuse the majority of viewers.
Even if you fine with most viewers being confused by the burn card (because that makes poker players "elite"), they would still have to change Vince's culminating line to: "If this card AFTER THE NEXT is a Jack of Spades, you all were trying to *beep* me!" Doesn't quite have the same ring.
So please just give it up:
-You are not better poker players than the writers
-You are not more detail-oriented than the writers
-You certainly are not better at writing television episodes
Sorry.