Stupidity: Considering he had a watch on in nearly every other episode, the Professor did not have his watch in this episode where they had a rescue deadline time (11:00) to be picked up along with the two Russians. Even if he forgot to put his watch on, he definitely would have paid attention to what time it was and got everyone to the location well before 11:00 to guarantee the rescue. Instead he relied on Gilligan to come up with the time which failed (an earlier submitted entry explains that part).
Stupidity: All castaways knew about the highly potent drinks, but they each drank it all instead of just pretending to drink with fake sips each time Lord Beasley took a sip - thus knocking every one of the castaways out cold and losing another rescue.
Stupidity: After Eva Grubb first arrives on the island and states her frustrations to the castaways, she walks away and the Professor reveals a plan to not leave her in her current condition and have a boat sent back for her to get her some help. How is he going to do this? If he could contact civilization to get Eva a boat, then how come he could never do the same to get themselves rescued? Furthermore, how would he be able to contact anyone to help Eva anyway?
Suggested correction: You completely misunderstood his plan and the episode up to that point. Eva arrived by boat and ended up giving the boat key to the group. The group was going to leave the island on Eva's boat, but she wanted to stay and wouldn't go with them. But in his opinion, she was too emotional to stay on the island alone, so after they got back home, using Eva's boat, the group would find help and that help would go back to the island to bring Eva home.
Smile, You're on Mars Camera - S2-E4
Stupidity: When the castaways stand in front of the camera waiting for it to operate, they hold signs up saying "SOS", "Help!", etc. thinking that will get them rescued. Even once the camera finally works and the gentlemen running the camera see them, this would do the castaways no good considering there is only picture (No sound) and the viewers think this is Mars. No location they were at were on the signs, and they can't be heard so the signs made were fully irrelevant for a rescue attempt.