# “Reasonable doubt” and conditional probabilities

Let A be the event that the accused is guilty and B is the event that evidence / testimonies are presented against the accused.

In the U.S., does "reasonble doubt" refers to a juror's subjective belief about the conditional probability P(A|B)? ("P" here refers to probability.) Or "reasonable doubt" refers to the unconditional probability P(A)?

Further clarification: with Bayes's rule:

P(A|B) = P(A and B) / P(B) = P(B|A)*P(A) / P(B).

In the U.S., which of the probabilities above (if any) do the prosecutor care about? What about the defense?

Edit for extra clarity:

I'm not saying how a juror actually thinks or should think about "reasonable doubt." I'm asking if reasonable doubt can be cast in this light.

• I'm not sure Law.SE is the right place for this question: the assumptions that 'reasonable doubt' can bear a Bayesian interpretation, and that anybody (let alone both prosecution and defence) care about probabilities, need support, to say the least. Furthermore P(B), as currently presented, is nearly meaningless. – Tim Lymington Oct 1 '18 at 17:48
• @TimLymington Suppose in a particular case, B is the event X accuses of Y a certain crime. And A is the event that Y actually committed the alleged crime. Then, P(B) is the (unconditional) probability that X accuses Y of said crime, which should be a pretty low number, and P(B|A) is the conditional probability that X makes the accusation given that the crime in fact happened. Supporters of X will surely argue that P(B|A) is high, or 1 itself. Anyway, I appreciate the comment. I don't claim to have figured this all out. Asking the question here is part of the process. – yurnero Oct 1 '18 at 17:53
• It may be that you are seeing "what is the probability of someone being charged?" as equivalent to "what is the probabiltiy of a fair coin coming down heads?" (which does indeed need to be included in a rigorous study). But actually it is more like "what is the probability of a coin being fair?" If that question has an answer, it is not a mathematical one. – Tim Lymington Oct 2 '18 at 21:41

In the US the jury is the fact-finder, while the judge is responsible for decisions purely related to the law. The defendant cannot be convicted unless the jury is convinced beyond a reasonable doubt that the defendant is guilty. Presumably the OP's probability P(A) is the probability the defendant is guilty given all information available to some ill-defined "public", which would presumably exclude secrets only known to the defendant, the defendant's lawyer, the defendant's priest, witnesses who never came forward, the undetected real perpetrator, etc.

I think the OP's intent is P(A|B) be the probability, as it ought to be evaluated by the jury, of guilt given the evidence and testimony presented to the jury during the trial. This often different than P(A) because some evidence is deemed to be inadmissible (for example, because police found it during a search, but they failed to obtain a warrant to perform the search) and withheld from the jury. It's also possible that a juror might be aware of evidence that the juror shouldn't have seen, such as a juror who improperly visited the scene while court is not in session.

The system cares about P(A|B), not P(A). Defendants are routinely found not guilty even though inadmissible evidence makes it clear they did it. Convictions have been overturned when it was learned that a juror improperly obtained knowledge of evidence that was not presented during the trial.

The phrase "reasonable doubt" was formed hundreds of years ago, and does not hold any mathematical or probabilistic meaning.

It is for each individual juror to decide for themselves what constitutes "reasonable doubt", and whether the evidence presented to them has crossed that threshold.

EDIT for extra clarity:

As stated above, the definition of "reasonable doubt" is intentionally vague*, and left to be decided by each juror for themselves on a case by case basis; as such there is no single rule that can be applied to jurors (also note barring accepting a bribe, a juror cannot be legally sanctioned for their conduct as a juror, nor their vote, regardless of the evidence before them).

So one juror might judge by P(A|B), another might judge by the defendant's appearance, another might judge by the majority of their peers (so that they can go to a ball game that evening, such as in the film 12 Angry Men), another might disagree with the law (see: jury nullification) and so vote not guilty on that basis, and another might bow to social pressure and convict despite overwhelming evidence that the defendant is not guilty (for example, at the end of To Kill A Mockingbird).

A prosecutor cares about convincing the entire jury that the defendant is guilty(outside of Oregon and Louisiana, where only 10/12 vote is needed to convict, so the prosecutor only cares convincing 10 jurors).

The defense only cares about convincing a single juror (or three in LA or OR), although more can be useful to prevent a mistrial. The defense (in theory) should not care whether or not the defendant is guilty.

*The origin of reasonable doubt was in Britain, where certain jurors would refuse to convict, despite any evidence, due to religious prohibitions of "Judge not, less ye be judged".

• I'm sorry but I don't think this answers the question. If a person says "I believe it will rain tomorrow", there is a probabilistic interpretation of this statement (esp. if you take the Bayesian view) regardless of whether the person making the statement intends that interpretation or even whether the person has any understanding of probability at all. – yurnero Sep 27 '18 at 19:15
• Actually, it answers the question perfectly fine. That you don't like it doesn't matter. – gnasher729 Sep 27 '18 at 19:51
• @gnasher729 That last bit seems more confrontational than necessary. I'm just trying to see if it is possible to think about "reasonable doubt" concept through a mathematical lens. Likewise, the presumption of innocence can be framed in a decision theory framework where the cost of one type of error (convicting an innocent) is deemed much higher than the cost of the other type of error (exonerating a guilty). It doesn't seem relevant whether the originators of "presumption of innocence" really thought about it using decision theory or not. – yurnero Sep 27 '18 at 21:36
• @yurnero If you're ever on a jury for a criminal trial, you can certainly think about reasonable doubt as a product of Bayesian probability. If you can get the data, you can try to do studies on the practice of reasonable doubt expressed in a formal manner. However, you can't get your ideas applied to jurors in general, many of whom have no idea of Bayesian probabilities. – David Thornley Sep 27 '18 at 22:58

The jury is supposed to base their decision on whether the evidence presented at trial has convinced them, beyond a reasonable doubt, that the defendant is guilty.

The mere fact that the defendant is being prosecuted (B) is not, in itself, evidence of guilt (A), and the jury is not supposed to consider it as such.

• Concerning the phrase: "whether the evidence presented at trial has convinced them, beyond a reasonable doubt, that the defendant is guilty", is a good understanding of this is: "based on" the evidence presented at trial, a juror is convinced or not. Isn't this then referring to the conditional probability P(A|B)? Maybe I can re-define my B better, but it seems that there is a conditioning event here. – yurnero Sep 27 '18 at 19:12
• I interpreted B as "the event that defendant is on trial". This is not to be considered. But if, for instance, evidence x,y,z has been presented at the trial, then they should consider the probability P(A|C) where C is the event that evidence x,y,z exists for/against the defendant. – Nate Eldredge Sep 27 '18 at 19:20
• Suppose for example that everyone in history who has ever been put on trial has turned out to be guilty. Then we would have to say P(A|B) = 1. However, if at this particular trial, no convincing evidence of B's guilt is actually offered, the jury must nonetheless acquit. – Nate Eldredge Sep 27 '18 at 19:23
• But if only P(A) were considered, then nobody would ever be convicted. That would be to say that the jury should ignore all the evidence and just consider the likelihood that a randomly selected member of the population is guilty of murder (say). That likelihood is very small, so they would acquit. – Nate Eldredge Sep 27 '18 at 19:24
• @yurnero I still don't understand the question. The probability that some evidence is presented against the defendant has no bearing on whether the defendant did the crime. For example, suppose someone was murdered in my apartment on Tuesday evening. Whether I was at home on Tuesday evening has some bearing on the probability that I committed the crime. The probability of that evidence being presented at trial does not. – phoog Oct 1 '18 at 17:10

This standard of proof means that, if we accuse someone of a crime ("Jimmy Jones Stole the Cookies From the Cookie Jar") and that person denies the claim (Mr. Jones says "Not me, Couldn't be!") that there are two stories that are in conflict, one must be true. Once I make the charge and present my evidence, I must show that my accusation is true. Jones on the other hand, needs to prove that it could be false.

Assuming a search of both the crime scene and the suspects property show evidence of access to the scene by prints on the front door and Jones' fingerprints on the lid, but none of the stash of missing cookies in Mr. Jones' property but we do find a key to the crime scene, we can show that Jones was at the crime scene... but that's because he's a good friend and I gave him the key to my place. So, we can assume he didn't steal the key.

Jones also points out that the police did find a partial print which matched neither myself nor Mr. Jones and I confirm that we are the only too people with keys to the house, this means that someone was inside the house who had not been there. Mr. Jones also points out that there is no finding of prints on the back door... but that's because my local CSI team was too underfunded to afford CSI of the level of Horatio Caine from Miami... so we got a guy who completely forgot to dust the back door for prints.

At this point, there is a reasonable doubt to Jimmy Jones stealing the missing cookies, as he did not have them, he was trusted by me and thus has little reason to steal from me... and there is an unidentified third person who was found in my home and I did not give anyone that access.

Because there is some evidence against my claim, the jury finds Jimmy Jones Not Guilty. It's important to note that legally this doesn't mean he didn't do it, but rather, I don't have any way of showing ONLY he didn't do it. Jimmy Jones' need not make a case of unidentified drug dealers who remain uncaught... only that my evidence does not fully refute his claim that there exists a possible second story.

In short, the Prosecution, once at trial, must show that their claims have enough evidence so that 12 random people (typical Jury rules... there are some states, where it's as low as 10 out of 12, but most are 12) can agree that this is the only possible way this could happen (100% probable). Where as the Defense, needs to show that there is the possibility for some probability of a second story. The Prosecution cannot be doubted at all.

There's any number of cases in real life cases that show that because of evidence. In real life, the OJ Simpson case went for this, some evidence of bias on the parts of the investigators, and some improper evidence storage (the famous gloves) that called the case into doubt. Also the crap treatment of the jury by both sides.

The real life Casey Anthony case is as close to a best example of a Real Life modern Rashamon as can be seen in a tried case.

The fictional events of the aforementioned Rashamon deal with the idea of who do you believe when the seemingly impossible course to reconcile events play out (Namely Three people all claim to be solely guilty to the same murder and their description of the events surrounding their guilt... including the murder victim!).

"My Cousin Vinny" is a great story about how objective material evidence is better than eyewitness testimony. That just because you are falsely accused of a crime, it does not mean that the prosecution's side is evil (almost all of the the typical "bad guys" in a legal drama are shown to be people who are doing their jobs and made honest, but hilarious mistakes). The definitive blow to the prosecution's case is a sight to behold... and the fact that everything is foreshadowed earlier makes it great on rewatch.

Twelve Angry Men shows how Jury Deliberations are really decided by 12 different perspectives having to agree on the conviction and how difficult it is to get a vote in either direction.

And then there is To Kill a Mockingbird which inspired untold numbers of American Trial Lawyers (especially defense lawyers) because of the obvious abuse of the system being permitted by the community writ large.

A conclusion of guilt is a deductive inference given a set of premises, all of which must have been judged to be true. For example, to conclude guilt, you must have concluded that the accused caused the death of Smith (and that it was not legally permissible to do so, etc.), so your A should be one of the elements of the crime (for all of the elements). Failing that, you have reasonable doubt. What you are testing is whether it is improbable that defendant did not kill Smith, while at the same time defendant was seen with Smith minutes earlier and defendant hated Smith and defendant had Smith's blood on his hands. The defense may wish to show that one of these premises is false, or that there is a missing premise (Jones entered the room just as defendant left).

• I see what you are saying. Your considerations expose the deficiency of thinking about the issue in terms of conditional probabilities. Either that or I haven't defined B very well. Suppose B is redefined as "the event of the existence of presented evidence / testimonies." Then, can "reasonable doubt" in this case be about a threshold for P(A | B)? And if a defense poke holes in the evidence / testimonies, then this can be translated as claiming P(B) = 0, and thus making the conditional probability ill-defined all together. – yurnero Sep 27 '18 at 21:45
• And if the defense introduces a new evidence, then we can redefine B as "the event of the existence of the newly presented evidence / testimonies." Put differently, we can think of the "presented evidence / testimonies" entailed in the definition of B is the final version of "presented evidence / testimonies", just before the jurors make the deliberation. – yurnero Sep 27 '18 at 21:47
• @yurnero but there will certainly be presented evidence and/or testimony. How is that probability relevant if it is certain? – phoog Oct 1 '18 at 17:13