I think the real issue that most people have understanding this stuff is that they don't have a clear grasp of dependent vs independent events. We tend to use

I think the real issue that most people have understanding this stuff is that they don't have a clear grasp of dependent vs independent events. We tend to use the fair coin or the fair die as a common way of talking about things, but I've found that this is problematic for a lot of people. First off is the use of the word fair. People not from a background in statistics don't think of that word the way we do. They think of it as an attribute of the object itself. The the coin or die is 'fair' in the ethical sense of the word. Second is the experiential understanding: 'This coin is fair and therefore should flip heads and tails with equal frequency. It's the same damn coin, and I flipped it a thousand times! Why are these crazy stats people telling me it comes up heads and tail equally often? It doesn't, dammit!' What I've had some success with helping people understand independent trials is to get them to envision flipping 6 different coins at the same time. Then you can sort of walk them back into understanding that flipping 6 different coins once is no different from flipping the same one 6 times. The events are totally unrelated, and a past event is not a predictor of the future. Plus, you can actually map out the math for them on the spot. .5 * .5 * .5, etc. That's just my experience trying to explain things though. Would love some criticisms if there are better ways to go about this.