Why should someone believe another person's assertions? I would imagine that the answer is because most of the times when I have been able to verify other people's statements, I've found them to be true.
Showing a position is true or probably true is impossible. The idea that it is possible and desirable to prove ideas true or probably true (justificationism) is wrong. In reality, you can't prove any position or show it is probable. Any argument requires premises and rules of inference and it doesn't prove (or make probable) those premises or rules of inference. If you're going to say they're self evident then you are acting in a dogmatic manner that will prevent you from spotting some mistakes. If you don't say they are self evident then you would have to prove those premises and rules of inference by another argument that would bring up a similar problem with respect to its premises and rules of inference.
In reality all knowledge is created by conjecture and criticism. You notice a problem with your current ideas, propose solutions, criticise the solutions until only one is left and then find a new problem.
A person should adopt a position as a result of it solving a problem and having no known criticisms. Whether the assertion comes from another person is not relevant.
See "Realism and the Aim of Science" by Karl Popper, Chapter I.
This becomes more likely as more people corroborate on a particular
No. The majority of people in the past had ideas that we now think are false. Either they are wrong or we are but in either case lots of people believing something has no bearing on its truth.
But can this be quantified in probabilistic terms (in other words, the
probability that someone is telling the truth) in order to balance
such a likelihood against other prior possibilities?
No. An idea is either true or false, it doesn't have a probability.
Furthermore, there are whole disciplines where I know that I'm not
qualified to validate _any_statements made by any of the experts in
that field, such as that of quantum mechanics, but trust the
'establishment' of the scientific community. While sometimes claims of
'the scientific community' are verifiable, there are whole areas
where, from my own perspective, they are not. How can I evaluate the
claims of such 'experts'?
Your qualifications are irrelevant. What matters is whether your ideas are right or wrong.
You have several options.
(1) Don't evaluate their claims at all. If you have no problem to which quantum mechanics is relevant then you don't need to have an opinion on it.
(2) Read stuff they write for popular audiences (e.g. - "The Fabric of Reality" by David Deutsch, Chapter 2) and see if the explanations they give make sense or not. If you think a particular position makes sense (i.e. - solves the problem it set out to solve and doesn't contradict other knowledge) you can adopt it.
(3) Learn quantum mechanics if you want more detail.