[csense]

It's obvious to me that writing "Give me a confidence score from 1-5" in your prompt will have disappointing results if you use that score directly. If you want self-reported confidence scores that are at all useful, you need to think about statistics and score calibration regardless of whether those scores are produced by an LLM or a human.

Take a corpus of problems you know the answers to, but the AI sometimes gets wrong [1]. Have the AI try to solve each problem and give you a 1-5 integer confidence score. The known proportion of correct answers in each bucket gives you a mapping from scores to probabilities [2].

[1] One possible corpus creation strategy might be math problems that you really need to run a program to solve. You can run the program to generate the correct answer key, but disallow tool calling for the AI.

[2] I'm pretty sure there's some stats wizardry that will let you put error bars on each bucket's probability based on the number of problems that end up in each bucket.