Reading the comments, I'm not sure we're addressing the question you're asking.
Prediction market proponents have put a lot of thought into how to turn mind-killing open-ended questions (far-mode) into resolvable bets (near-mode), often by use of conditional wagers (chance of Y, wager valid only if X happens). This is a good mechanism for applying percentages to such questions.
However, it completely bypasses the point I think you're trying to make, which is that all useful beliefs are of this form. Basically, you want a simple-example summary of Making Beliefs Pay Rent.
I'm not sure there exists such a single example - humans are very good at separating the signaling and identity "beliefs" from testable, resolvable "predictions", and each of us has different topics with different ease of analysis. This means that it takes quite a bit of thought, study, and reflection to see how the word means different things, which shouldn't be mixed.
I think the direction to go in these discussions isn't so much to find a topc that's often expressed as an identity-signal "belief" and push someone toward a prediction-probability-"belief". Instead, you can ask "what choices are you making based on this belief, and what's the cost if you're wrong"? When you find topics on which the individual can admit that there is a concrete wager being made, then you can guide them to understanding how belief works.
And of course, if they can't tie their belief to any decisions, or understand that the decisions would be different if the belief were incorrect, then you can point out that this kind of belief doesn't help the holder of the belief.
When I explain to people how beliefs should be expressed in probabilities, I would like to use an example like "Consider X. Lots of intelligent people believe X, but lots of equally intelligent people believe not-X. It would be ridiculous to say you are 100% sure either way, so even if you have a strong opinion about X, you should express your belief as a probability."
Trouble is, I'm having a hard time thinking of an example to plug into X. For an example to work, it would need the following properties:
Factual question. So no value-laden questions like "Is abortion morally acceptable?" or counterfactual questions like "Would the US have won a war with the Soviet Union in 1960?"
Popular and important question. The average person should be aware it's an issue and care about the answer. So no "My aunt's middle name is Gladys" or "P = NP."
High uncertainty. Reasonable people should be reluctant to give a probability >90% or <10%
No opportunity to gain status by signaling overwhelming support for one side. So cryonics is out, because it's too easy to say "That's stupid, I'm 100% sure cryonics won't work and no intelligent person could believe it." I'm assuming in any debate where you can gain free status by assigning crazy high probabilities to the "responsible" position, people will do just that - so no psi effects, Kennedy assassination, or anything in that cluster. I need a question with no well-accepted "responsible" position.
Minimal mindkilling effect. My previous go-to example has been global warming, but I keep encountering people who say that global warming 100% for sure exists and the only people who could possibly doubt it are oil company shills. Or if I were to try the existence of God, I predict half the population would say it's 100% certain God exists, and the other half would say the opposite.
So what are the important questions that average (or somewhat-above-average) people will likely agree are complicated open questions where both sides have good points? And if there aren't many such questions, what does that say about us?