It's an interesting question, if someone is right for the wrong reasons, are they still right?
Logically, it's a no-brainer - right is right, for whatever reason. But not every opinion is a logical one only - there are things like moral positions that cannot be called merely logically or factually correct, simply conventionally correct, moral, right, even if irrational. So if someone expresses a factual truth based on irrational reasons, they may be factually correct, but we may feel morally obliged to discount their opinions as having any weight.
So the problem is with the word "right". It sounds moral. Logicians use "valid" and "invalid" in logical reasoning, and "true" or "false" in truth tables. That is, conclusions deriving from premises (or terms) may be valid or invalid deductions, or true or false conclusions, so that while an invalid deduction may be factually correct, the premises were wrong (your argument), and therefore the "argument" is invalid. We just have to know what the argument - the reasoning - is. If someone says "I don't believe that Tarot was based on Hebrew because I hate Hebrew, so I vote 'No'", then the reasoning is pretty weak, so weak that we can characterize it as wrong - the argument is invalid.
The broken clock example is still pretty useful - if you didn't know that a clock was broken at 6 o'clock, and happened to be wondering what time it was
at 6 o'clock, and you glance over at the clock you don't know is broken as you pass by, you will get the correct time. But it was pure accident, coincidence - the clock is right, and you get the right time, but for the wrong reasons. Yes, it is 6 o'clock, but no, the clock is not telling the right time, because it is not telling anything - it's stopped. Coincidence caused the illusion of a correct result.
Try the reverse method to see if it casts any light on the positive formulation -
If someone is wrong for the right reasons, are they still wrong?
It irks that way, doesn't it? If someone has right reasoning, surely their thinking will be right.
So how can someone with wrong reasoning have a right opinion?
Let's use the clock example again. Let's say a clock is working just fine, it's just telling the wrong time. But you don't know that, so you are late for an appointment. You were wrong about the time, but for the right reasons - you didn't know the clock was wrong.
Does it work morally? It can't really, because moral rightness and wrongness are arbitrary, however firmly we may believe otherwise. But we may say that the man who missed his appointment had a moral obligation to make sure his clock was right, and that therefore his reaoning - trust in this one clock - was wrong. So the negative formulation doesn't work if we apply moral considerations to the factual ones - he was wrong, and had wrong reasons as well.
So back to the positive formulation. If someone is factually right by accident, they remain factually right, but we may feel that, like with a broken clock, that the opinion is not to be trusted. That is, there is a
qualitative dimension to the "truth" in this instance, not merely an absolute right or wrong. That quality is what I call the moral dimension.
So the only way to test the quality of our poll results is for each voter to give their reasons. We can only accept those that display a mastery of the relevant data. So - how do we decide what is objectively relevant?
Yes, back to authority, as is the case in all moral questions.