Is there a legal system or theory where punishment is proportional to the probability of an offence?

Question

In common criminal law in order for a defendant to be punished, they must be found guilty beyond all reasonable doubt. Hence, there is a threshold for the probability of guilt, beyond which the punishment is proportional to the offence and/or consequences.I want to know if there is a legal system where punishment is proportional to the probability of guilt. For example, Alice and Bob are equally likely to have committed a known crime and one of them must have committed the crime. Hence, each faces 50% of the normal penalty. Such a legal system could be from anywhere in the world at any point in time, from modern roman legal systems, to small systems governing tribal communities. Any of these are interesting to me if researched.I am also interested if such a punishment system has been investigated theoretically. Primarily, what the benefits are, but the ethical side is also of interest.If anyone can help that would be most helpful, I have not been able to find any scholarly or other types of article. I am interested from a mathematical perspective, but I'd like to know whether such systems exist in reality.UPDATEJust to be clear, I am not asking whether there has been implemented a mathematically precise decision procedure, weighting level of punishment proportional to probability of guilt. But rather, if there is or has ever been an informal (i.e., imprecise) system.There are good reasons for there not to be, I am asking whether there has been or is such a system.Thanks for the answers so far, I just wanted to clarify that.

Answer 1

There was an Arizona case, prosecuted against 2 codefendants, separately. In the 1st trial, the prosecuted argued that guy was the trigger man, while the other guy was just an accomplice. The jury convicted. Then the prosecutor went after the 2nd guy, and argued again that he was the trigger man, the real killer, and the 1st guy just an accomplice. The jury convicted again.

Trials are crap shoots, roll of the dice with prosecutors doing somewhat unethical things, as there is no serious sanction for their concealing evidence or twisting facts. So, there is little correspondence between probability of guilt and sentence, after a trial. It's all or nothing. The way it does get into the mix is plea bargaining. The weaker the case, the less probability of guilt, the more lenient the plea deal. Convince the prosecutor he may lose, and the plea agreement gets more lenient.