Margin of error

I've been looking at political polls a lot recently. In my attempt to understand them, I've been thinking about the terms thrown about with explanation, one of which is "margin of error".

I don't think margin of error means what people think it means. You will often see polls like this: Bruce Scott (R) 51% v. Scott Bruce (D) 49%, margin of error 3%. To which people respond, "that's a dead heat since it's within the margin of error". Not exactly. Margin of error cuts both way; the real percentage is as likely to be 54-36 as it is 48-52. Given that, there are seven possibilities (54-36, 53-37, 52-48, 51-49, 50-50, 49-51, and 48-52). If one assumes that each of these possibilities is equally likely, then the "republican" wins 4 out of 7 times, the "democrat" 2 out of 7 times, and the lawyers 1 out of 7 times.

I realize that this is a gross oversimplification. To begin with, votes almost never break in even integer percentages, but the 2:1 win ratio for the "republican" v. "democrat" will apply for fractional results as well, the percentage of ties, however, will shrink. It is also probably that there is some sort of bell curve for these possibilities, so that 54-36 and 48-52 are equally likely, but both are less likely than 53-37 and 49-51. Even given that, the area under "republican" side of the curve will be greater that the area under the "democrat" side of the curve. I suspect that the 2:1 ratio might even still apply.

I also realize that I've never taken a course in statistics or probability, and that I may be completely wrong here. But until someone explains to me why this reasoning is wrong, I'm going to assume that "margin of error" is something that can almost be ignored.