Between teaching mathematics to freshmen and spending most of my time learning mathematics, I’ve noticed this myself. When presented with a new result, the first inclination, especially depending on the authority of the source, is to believe it and figure there’s a valid proof of it. But occasionally the teacher realizes that they made a mistake and may even scold the students for not noticing since it is incredibly obvious (e.g. changing something like ||z—z_0|| to ||z—z_1|| between steps, even though a few seconds thinking reveals it to be a typo rather than a mathematical insight).
Sometimes (and for a few lucky people, most of the time) individuals are in a mental state where they are actively thinking through everything being presented to them. For me, this happens a few times a semester in class, and almost always during meetings with my advisor. And occasionally I have a student who does it when I’m teaching. But in my experience this is a mentally exhausting task and often leaves you think-dead for a while afterwards (I find I can go about 40 minutes before I give out).
All this leads me to a conclusion, largely from my experience with what behavior produces what effects, that in mathematics the best way to teach is to assign problems and give students clues when they get stuck. The problems assigned, of course, should be ones that result in the student building up the mathematical theory. It’s certainly more time consuming, but in the end more rewarding, in terms of both emotional satisfaction and understanding.
zzz, I think you underestimate how people perceive gambles. Investing in financial markets isn’t perceived as a bet, since we like to believe that if you only knew enough, you could make the right choices (whether you actually can or not is another matter). With lotteries and other forms of gambling, it doesn’t matter how much you know, you can’t anticipate the outcome any better than if you had no additional information. That, I think, is part of why gambling is much more popular than investment: even the least skilled person has the same chance of winning as the most.