I'm not sure that I see your problem. Although Probability is supposedly used in Statistics, which is the subject of this article, they are not one and the same. Care to clarify?

And no, we have not one bit of mercy.

When saying this sample has 3% expected error, what you actually mean is:

The statistical analysis of this sample has X precent chance of being withing 3% range of the right answer.

Probability comes in when you calculate how large the sample should be to have a given chance of being withing a given error. Or what is the chance of being withing a given error with the sample size you have.

Hmm...
Couldn't they mean that there is a 3% chance of the real result being outside a single standard deviation radius from the mean? Or something completely different? They have, indeed, not given us any notion of what it is that they consider an error.
Could they mean that the standard deviation is 3% of the votes?

Problem is, that when you are referring to elections, there are at least two percentages you can be referrring to: those regarding probability, and those regarding the actual percentage of public support each candidate had received.