# Probabilistic justice or what is reasonable doubt

We all know that a person should be convicted of a crime if there is no reasonable doubt that the person is guilty.

Let's consider this ideal case.

Bill was killed. Police suspects John. John says, "It is not me." They go to court and there is only 1 evidence some kind of expert report, which tells that probability that Bill was killed by John is 80%. Or maybe this report is a summary based on many evidence, but anyway, final result of expert conclusion "probability that Bill was killed by John is 80%."

Let's say expert is very professional, experienced and nobody has doubt about his competence and etc. Everybody trust this expert.

What is the correct verdict in this case?

If 80% is not enough to establish guilt, what should the number be? Is it 90%, 95% or 99%?

UPDATE:

Or for example pure mathematical case. Maybe it is stupid example, because I do not expert in all these things, but still. On shooting range was 3 persons, John, Bill and James. John and James shot from the gun at the same time and Billed was killed accidentally, unintentionally. They had the same guns and bullet was not found and they cannot determine exactly who's bullet actually killed the Bill. Let's assume that it was proven, that for both John and James it was criminal negligence. Somebody definitely have to go to jail. But some ballistic experts calculated bullets trajectory and mathematically proved that probability that it was John's bullet 80%, probability that it was James bullet was 20%. Who is going to jail? John or James? And interesting what will happen if calculation gives 50% and 50% :)

UPDATE2:

Another example, almost real :) Beginning the same as initial story with killed Bill and suspected John who declines everything. There is DNA analysis and something went wrong, for example there was not enough DNA or it was partially damaged and expert gives some numbers, that DNA of killer match with John's DNA with probability 80% (or 90% or 95%).

Please before giving long explanation answer briefly, what by your opinion and from your experience court will decide in the cases UPDATE and UPDATE2? Just something like 80% - not guilty 90% - not guilty 95% - guilty

or less 99% - not guilty, more than 99% guilty.

I do not ask how it is defined in the law, it is clear that in the law there is no any numbers like this. I ask what should be threshold by your opinion and from your experience?

## Beyond Reasonable Doubt

In Australia, the High Court has decided in Green v R:

A reasonable doubt is a doubt which the particular jury entertain in the circumstances. Jurymen themselves set the standard of what is reasonable in the circumstances.

And they further said that judges should:

adhere to and not to attempt needless explanations of the classical statement of the nature of the onus of proof resting on the Crown.

That is: "beyond reasonable doubt" means what that particular jury decides that it means and any judge who tries to clarify it beyond that is setting up an appeal for whichever side loses.

This is reflected in the NSW Supreme Court's Criminal Trials Bench Book (essentially a crib sheet for judges):

Let me now say something to you about the onus of proof. This is, as you have already been told more than once, a criminal trial of a most serious nature and the burden of proof of guilt of the accused is placed on the Crown. That onus rests upon the Crown in respect of every element of the charges. There is no onus of proof on the accused at all. It is not for the accused to prove [his/her/their] innocence but for the Crown to prove [his/her/their] guilt and to prove it beyond reasonable doubt.

It is, and always has been, a critical part of our system of justice that persons tried in this court are presumed to be innocent, unless and until they are proved guilty beyond reasonable doubt. This is known as the “presumption of innocence”. This expression “proved beyond reasonable doubt” is an ancient one. It has been deeply ingrained in the criminal law of this State for almost two hundred years and it needs no explanation from trial judges.

The Crown does not have to prove, however, every single fact in the case beyond reasonable doubt. The onus which rests upon the Crown is to prove the elements of the charges beyond reasonable doubt and I shall subsequently outline to you the elements of the charges.

In a criminal trial there is only one ultimate issue. Has the Crown proved the guilt of the accused beyond reasonable doubt? If the answer is “Yes”, the appropriate verdict is “Guilty”. If the answer is “No”, the verdict must be “Not guilty”.

The jury in each particular case will consider the evidence that has been put forward and either accept it as true or not true (or some of it as true and some of it as not true). Having decided what evidence to accept that jury needs to decide if the state has proven each element of the crime beyond reasonable doubt and that jury decides what that phrase means. That is their job!

• Thanks for the answer. If you would be juror, what you would decide in my examples? In ideal case if there are no any other evidence, but only this report with percentage? Commented Nov 4, 2016 at 13:50
• Your hypotheticals are so absurd that the question of what I would do is meaningless - a better question is what I would do in a zombie apocalypse- that's way more likely
– Dale M
Commented Nov 4, 2016 at 21:31
• @Zlelik your strawman case also doesn't specify WHAT CHARGE. If someone is killed at a shooting range the charge could range from first degree murder to willful negligent discharge of a firearm. There are many more elements to the first charge than the latter. And the last charge could be a Felony or a Misdemeanor! Commented Oct 10, 2023 at 17:54
• I put there, that it was accidentally, unintentionally and it was proven, that for both John and James it was criminal negligence. What else has to be clarified? Even if the charge is different, I think it does not matter. Whether it is 1 year in jail or 10 years in jail does not matter, the question is who goes to jail? Or I do not understand something? Anyway, you are free to give an open answer like this: - if it is first degree murder, then answer is this - if it is willful negligent, then answer is this if there are 3-5 options, it should be fine unless there are 100 possibilities. Commented Oct 11, 2023 at 19:33

Zlelik it's not a percentage. It's an absolute bar on a guilty verdict if there exists at least one doubt that is reasonable.

• Do you mean 100%? In this case question is what is reasonable "doubt"? In case with DNA "there is a 1 in a trillion chance that the match is accidental" there is small probability (0,0000000001%) that DNA match accidentally. is it a reasonable doubt or not? Anyway, what you would decide, if you get a report with numbers 80%, 90%, 95% and etc? In case of DNA example it would be 99,9999999999% guilty. Commented Nov 3, 2016 at 22:10
• @Zlelik "reasonable" is largely subjective. It is up to the juror. Maybe the accused has a genetically identical twin who was separated at birth. Is it reasonable to suspect that? Commented Nov 4, 2016 at 6:47
• @phoog I do not know what is reasonable, this is why I asked this question. What if it really was his identical twin? Jury will send innocent person to jail? It is not right. From other point of view, if we accept that without reasonable doubt means 100%, we cannot literally prove anything. There is always small probability that person is telling truth, even if he tells, that Aliens came and killed that guy and put his DNA to dead body. I think if scientists still try to find life on other planets, probability that Aliens could come is not zero. What you would decide in my examples? Commented Nov 4, 2016 at 13:57
• @Zlelik I am sure that DNA evidence could -- and probably would -- send an defendant to jail if the defendant were not known to have a lost identical twin, even if the defendant were innocent and the identical twin were the culprit. Criminal justice system is far from perfect, and convictions that "are not right" happen every day, for much more mundane (and, perhaps, more reasonable) reasons. You say you do not know what reasonable is, but if you were to serve on a jury, you would be called upon to decide one way or another for one specific case. Commented Nov 4, 2016 at 15:11
• I doubt any jury would find a space alien defense to be reasonable. It's well established that "beyond a reasonable doubt" does not mean "with complete certainty" or "beyond a shadow of a doubt." The point in response to your question is that it is a matter of subjective judgment. There's no way to identify a threshold probability that defines reasonable doubt. For that matter, putting a number on the probability of a story's being correct is also subjective. Can anyone put a number on the probability that any given person has a long-lost identical twin? Commented Nov 4, 2016 at 15:17

The generic description of the burden of proof is that the state must prove the elements of the crime to the point that there is no reasonable doubt that the person is guilty. (Generally, it is juries who decide whether the case has been made, not judges). This matter has been well studied and it has been known for some time that jury instructions can significantly mislead jurors as to the intended standard of proof. See L. Solan Refocusing the Burden of Proof in Criminal Cases: Some Doubt About Reasonable Doubt.

One of the points that these studies show is that that the "classical" explanation of the concept suggests that the defense has to create a doubt – which is not correct. If the state provides a pitiful argument for guilt, the defense is not supposed to have to do anything; the case should fall on its lack of merit, but in fact older wordings of the instruction can leave jurors expecting the defense to now create a doubt.

The best instructions focus on the government's obligation to leave you "firmly convinced" of the truth of the accusation. Here are the instructions for California (well, ca. 2008):

Proof beyond a reasonable doubt is proof that leaves you with an abiding conviction that the charge is true. The evidence need not eliminate all possible doubt because everything in life is open to some possible or imaginary doubt. In deciding whether the People have proved their case beyond a reasonable doubt, you must impartially compare and consider all the evidence that was received throughout the entire trial. Unless the evidence proves the defendant guilty beyond a reasonable doubt, he is entitled to an acquittal and you must find him not guilty.

An expert witness might say that there is an 80% chance that John killed Bill, but the defense would attack that claim, and it is questionable that his specific claim would hold up, since he has no foundation for making a numeric claim. Such a probability does have a real basis if it is about DNA, because you can actually say things like "there is a 1 in a trillion chance that the match is accidental". Spock on Star Trek would often say that there was a 27.913% chance of a certain outcome, and that is just science fiction.

As far as I can tell, we don't actually know how jurors would mathematize their task, e.g., "Guilty means at least 90% probability of truth." I maintain that probability theory is not an appropriate tool to bring to the discussion. Instead, the question should be based on the nature of the reasons given for the conclusion. If a person testifies that he saw John shoot the victim, you decide if the person is speaking truthfully. His squirmy demeanor on the stand could indicate that he's hiding something; his known hatred for John could confirm the conclusion that he's lying, which could neutralize the value of that testimony. But maybe he has painful hemorrhoids and he is actually a good friend of John's.

The article whose results I referred to in the comment is "Burdens of Proof: Degrees of Belief, Quanta of Evidence, or Constitutional Guarantees", C.M.A. McCauliff, in Vanderbilt Law Review 35:1293–1335. However, I can't find a free copy online, but interlibrary loan might work. The author surveyed judge's answers to various questions, aimed at saying what the perceived numeric values are for "beyond reasonable doubt" or "clear and convincing evidence" (etc). Some similar studies are here, here, and this blog.

• Thanks a lot. But if you would be the juror, how you personally would define where is the border of reasonable doubt? 80%, 90%? Also what will happen in pure mathematical example, which I put in the question as update? Commented Nov 3, 2016 at 0:01
• It's not mathematical. It means that any doubts that you might have would be unreasonable. Commented Nov 3, 2016 at 8:54
• @gnasher729 I understand that in real cases decision is not based on mathematics. I ask what if there is ideal mathematical case where expert gives probability in number, what you will decide or what you think jurors/judge will decide in this case? Another example in UPDATE2. What you or jurors/judge will decide in this case? Commented Nov 3, 2016 at 22:23
• @Zlelik, both sides get to present expert witnesses. The mathematical claim simply would not survive. I suspect that as a matter of law, DNA evidence with so little match as you propose would be excluded as unreliable. Commented Nov 3, 2016 at 22:56
• Do you mean by this "DNA evidence with so little match as you propose would be excluded as unreliable" if DNA report tells about 80% probability match would be excluded? Which number by your opinion should not be excluded and means guilty 95% or 99%? Commented Nov 4, 2016 at 13:48