Plot hole: The crew makes no attempt to re-board the Enterprise or stop it from leaving the starbase even after they find out there is no danger from the anti-matter.
Plot hole: During evacuation, when Wesley is beamed off the ship, there are room for one more person on the transportation pad. You would think they would like to transport as many people away as fast as possible.
Plot hole: The entire premise of the episode is contradictory; the franchise makes abundantly clear through the so called Prime Directive that the Federation abides to a code (often creating moral dilemmas that may require to stretch the rules) that says that their staff is not supposed to interfere in world that haven't reached warp capabilities, nor involve themselves in their internal matters. In the first half of the episode, Riker and the others just go 'mingle' with the frisky natives without a care in the world, and yet in the second half the Prime Directive itself is referenced explicitly and it is part of the plot.
Suggested correction: They are just enjoying the hospitality of the planet's inhabitants. They are not interfering in their internal affairs or the development of their species. It's only when Wesley gets into trouble that the Prime Directive comes into play. The entire mission of The Enterprise is to make contact with other planets.
Even though they are supposed to make contact with other planets, it's pretty clear in the rest of the series after the first few episodes of Season 1 that they do not (intentionally) contact people that have no warp capability.
Plot hole: The binars needed two people to upload the data back to their world but they only intended on keeping Riker in the holodeck. Riker is just one person. Picard came into the holodeck purely by accident - even Minuet said this was true. If Picard hadn't been there, Riker could not have uploaded the data on his own and the binars' plan would have failed.
Suggested correction: The Binars only trapped Riker because they thought they might need someone to restore the computer for them, which wouldn't have taken 2 Enterprise crew members. It was the fact that the Binars got to their planet too late that required Picard and Riker to work together. However, the Binars already knew Picard was on the Holodeck before they initiated their plan because when Picard exits his ready room he says he's going to join Riker. Had Picard not done this, certainly the Binars could have kept a 2nd crew member on-board somehow, if needed.
