The logical side (or at least what I'm referring to), is the psychology side of it. How do you know that they aren't just asking you the question to try and change your answer so that you don't get the prize it's statistically the better choice?
The reason people "fail" is because they think it would then be a 50/50 statistically when it's much more complex than that. If they say the same one it may be because they might be thinking about it on a more psychological level rather than a mathematical level though so the "correct" answer really depends on the thought process.
I'm afraid I don't really understand where you're going with you comment on "being exposed to many doors". I can speculate based on your last sentence, but I don't really understand the relation very much. The reason that you would come to different logical conclusions about society issues when observed in small numbers is because there would be less diversity and interpretations, but of course would also be deeper and more meaning as well as more managable. If you are discussing society issue with a group of 3 other people, you are able to more easily go into further depth on there understandings of the problems. The down side to that, is that you would only go into their understandings which may lead to some bias. This is why I think that as far as with ethics and morality, that no one should be excluded because it would provide a more accurate scope as to what's actually ethical rather than what people what you to believe is ethical.
(This is a topic flashback that's just extra, I felt it made my main post too long).
That why i don't agree with that video you posted about Sam Harris and him think that only certain people who have "moral expertise". There's really no moral expertise there's really just expertise in how you present those idea's and the logic behind them. There are people in the world that are so persuasive that they could justify just about anything as being "moral". Once that happens it no longer becomes about morals really but about justifications and the more persuasive person wins regardless of whether they or their ideas are morally right or wrong. History has proven that murder, slavery, rape, torture, and so on, has been morally justified.