Twelve Angry Men and the Cumulative Weight of Evidence:

Twelve Angry Men, the classic movie discussed in David's post was a great film. But I've always thought that the defendant Henry Fonda's character persuades the jury to acquit was actually guilty. There were four or five separate pieces of evidence pointing to the defendant's guilt, including two separate eyewitnesses. Fonda's character does a good job of showing that no one piece of evidence was enough to prove the accused guilty beyond a reasonable doubt by itself. But he and the other jurors ignore the possibility that guilt might be established by the cumulative weight of multiple pieces of evidence that are individually insufficient. For example, let's assume that the defendant has five pieces of evidence against him and each of them individually shows that there is only a 70% chance of his being guilty. The combined probability of his guilt based on all five items is about 99.8%, more than enough to prove guilt beyond a reasonable doubt. By focusing on each piece of evidence individually, Fonda's character obscures this fact and persuades the jury to let a guilty man go. Obviously, this interpretation of the movie is not the one that the filmmakers wanted the audience to come away with. But I think it fits the evidence nonetheless.

In real life, this "divide and conquer" strategy was effectively used by O.J. Simpson's defense lawyers, who raised doubts about some of the individual items of evidence against their client, but successfully avoided confronting the fact that he was almost certainly guilty based on the cumulative weight of many different items of evidence. Former prosecutor Vincent Bugliosi discusses this in his excellent book on the Simpson case.

The point is applicable to issues beyond criminal law. People often dismiss individual arguments and evidence against their preferred position without considering the cumulative weight of the other side's points. It's a very easy fallacy to fall into. But the beginning of wisdom is to at least be aware of the problem.

UPDATE: My analysis assumes that the five pieces of evidence were conditionally independent of each other (i.e. - that the discrediting of one does not affect the odds of the others being valid). That, I think, is an accurate representation of the evidence in the movie, which consistent of several independent items: two separate eyewitness accounts, some items of physical evidence, and flaws in the defendant's alibi.

Related Posts (on one page):

  1. 12AM:
  2. Twelve Angry Men and the Cumulative Weight of Evidence:
  3. A Debate on "Twelve Angry Men":
CDU (mail) (www):
For example, let's assume that the defendant has five pieces of evidence against him and each of them individually shows that there is only a 70% chance of his being guilty. The combined probability of his guilt based on all five items is about 99.8%, more than enough to prove guilt beyond a reasonable doubt.


That's only true if the probabilities are independent, which may or may not be the case in a criminal trial (it depends heavily on what kind of evidence we're talking about).
2.1.2008 10:32pm
Ilya Somin:
That's only true if the probabilities are independent, which may or may not be the case in a criminal trial (it depends heavily on what kind of evidence we're talking about).

Yes, I agree. However, in the movie the items are all independent of each other: 2 completely separate eyewitness accounts, a couple of pieces of physical evidence, and flaws in the defendant's alibi.
2.1.2008 10:35pm
RodundaBall (mail):
For example, let's assume that the defendant has five pieces of evidence against him and each of them individually shows that there is only a 70% chance of his being guilty. The combined probability of his guilt based on all five items is about 99.8%

To avoid similar childish errors in the future, you might want to educate yourself about the interpretation of scientific evidence. Google "sensitivity specificity predictive value"
2.1.2008 10:48pm
BRM:
Sounds like a pretty good strategy for a defense attorney. Makes perfect sense to win the jury over by convincing the jury that the prosecution just can't be trusted to present accurate evidence.
2.1.2008 10:55pm
frankcross (mail):
I haven't even seen the movie, and so you may be right on the particular example, but this does suggest a common misuse of statistics often used in DNA testing cases. A 99.8% probability for an individual is roughly 2 in 1000. But if there are ten thousand people in the relevant sample, that means that 20 people will have that probability or higher. Which in turn means that the probability that this one individual did it is only 5%, not 99.8%. So this isn't really the right standard.
2.1.2008 11:05pm
Randy R. (mail):
Well, I suppose that's one way to look at it. Or you could say that each piece of evidence shows a probability that he is 30% innocent. The combined probablity of all five pieces would be higher than 50%, which should be enough to be reasonable doubt.

And then, if he is indeed innocent, there is all that absent evidence -- If he wasn't there, then there is no evidence for him to leave. That would make all that absent evidence a probablity of 100% innocence, which should be enough for acquital (except maybe Texas). right?

Of course, this also ignores the fact that the prosecution has the burden of proving all the elements of the crime, which it argueably did not in the movie.
2.1.2008 11:09pm
Ex-Fed (mail) (www):
In my experience on both sides of the v., evidence is rarely independent in any way that a moderately good defense attorney can't get around.

Moreover considering ALL evidence cumulatively vs. item by item isn't the only issue. Jurors (like people in many contexts) often engage in a manner of thinking that has a lengthy psychological name that escapes me. In short, they often focus on a single item or a couple of items to the exclusion of others, giving unjustified weight to those items. Hence the tried and true cockroach in the spaghetti argument.

"Ladies and gentlemen, the government tells you to ignore bad fact X and look at the rest of the evidence. They want you to ignore X. But I ask you, is it reasonable to ignore a fact as terrible as X? Is that how you act in the rest of your life? Imagine that you are at a restaurant and order spaghetti, and when the spaghetti arrives you find a cockroach in the middle of the plate. Do you carefully eat around the cockroach? Or do you send the plate back to the kitchen and say "this is no good?" Ladies and gentlemen, today the government is asking you to eat around the cockroach."
2.1.2008 11:25pm
Lake (mail):
I wouldn't really agree with this. It wasn't that the evidence implied guilt, but that it was entirely discredited. The old man couldn't possibly hear the murder, because there was a train going by at that exact time. The woman couldn't see it, because she didn't have her glasses on.

And the rest of the evidence was weak, anyway, and circumstantial. So what if he didn't name the movies, or he had a similar knife?
2.1.2008 11:28pm
Ilya Somin:
Well, I suppose that's one way to look at it. Or you could say that each piece of evidence shows a probability that he is 30% innocent.

No, the evidence I mentioned was not of that nature. None of it was potential proof of his innocence (e.g. - evidence that someone else did it, or that he wasn't at the scene of the crime). The only point at issue was whether it proved his guilt).
2.1.2008 11:30pm
Jagermeister:

Hence the tried and true cockroach in the spaghetti argument.
ROTFLMAO! I'll have to remember that one. A truly outstanding presentation.
2.1.2008 11:30pm
Ilya Somin:
It wasn't that the evidence implied guilt, but that it was entirely discredited. The old man couldn't possibly hear the murder, because there was a train going by at that exact time. The woman couldn't see it, because she didn't have her glasses on.

I don't think it was "entirely discredited." Background noise reduces the chance of hearing the murder, but doesn't eliminate it entirely. People with impaired vision can still see some things when not wearing their glasses. Moreover, it's highly unlikely that two separate eyewitnesses would just happenly to mistakenly conclude that they saw (or heard him) if he really wasn't there.
2.1.2008 11:33pm
CDU (mail) (www):
Well, I suppose that's one way to look at it. Or you could say that each piece of evidence shows a probability that he is 30% innocent. The combined probablity of all five pieces would be higher than 50%, which should be enough to be reasonable doubt.


I think this depends on the evidence. If the evidence is an eyewitness who is 70% sure, then the other 30% isn't necessarily innocence. An eyewitness who can't identify the defendant because she wasn't wearing her glasses isn't proof of the defendant's innocence, it's just not proof of guilt. On the other hand, there are some types of evidence where that 30% might be evidence of innocence. For instance if an eyewitness is 70% sure it's the defendant and 30% sure it's his brother, than the 30% would be evidence of innocence.
2.1.2008 11:35pm
CDU (mail) (www):
Moreover, it's highly unlikely that two separate eyewitnesses would just happenly to mistakenly conclude that they saw (or heard him) if he really wasn't there.


It depends on the circumstances. If one eyewitness sees someone at the time of the crime, then sees the defendant's picture in the paper the next day in a story saying he was arrested for a crime (based on the second eyewitness or other evidence), then it's quite possible that the witness will assume that the person she saw was the defendant, leading her to testify as such. Of course, this is also an example of why the probabilities of different pieces of evidence may not be independent.
2.1.2008 11:38pm
Ilya Somin:
Jurors (like people in many contexts) often engage in a manner of thinking that has a lengthy psychological name that escapes me. In short, they often focus on a single item or a couple of items to the exclusion of others, giving unjustified weight to those items. Hence the tried and true cockroach in the spaghetti argument.

This problem is known as the "availability heuristic." It is a real problem in many contexts. But I don't think it played a major role in this case.
2.1.2008 11:39pm
Daryl Herbert (www):
The alternative fallacy is to fall for someone else's baloney because they have a dozen pieces of "evidence," none of which stand up to scrutiny if one takes the time to scrutinize them.

Which is a huckster more likely to rely on: making up his own baloney, or by scrutinizing the other side's evidence and subjecting it to rigorous logic?
2.1.2008 11:58pm
Aeon J. Skoble (mail):
"it's highly unlikely that two separate eyewitnesses would just happenly to mistakenly conclude that they saw (or heard him) if he really wasn't there"

IANAL, but this seems especially weak. As speculated by several jurors, the witnesses had some reason to say what they thought the authorities wanted to hear. Given the actual narrative in the film, it's actually highly likely that both witnesses were mistaken. That's reasonable doubt.

IANA statistician either, but it seems wrong to argue that lots of unconvincing evidence somehow adds up to convincing evidence.
2.2.2008 12:00am
Randy R. (mail):
I believe all those child abuse cases must have been true, then, because all the children told stories that were consistent with what the authorities wanted to hear. Furthermore, they all named the same guilty parties.

Sorry, but eye witnesses testimony has been proven again and again to be the LEAST reliable evidence in any trial.

Why do I get the feeling that some people want to lower the burden of proof and make is that as long as there is some evidence of guilt, that's enough for conviction? I guess you could get rid of juries whenever you have two witnesses who agree on anything, because the probability of them being wrong is so low, we'll just skip the trial part and move right on to sentencing.
2.2.2008 12:14am
Nathan_M (mail):
I really wished I remembered this movie better. How much of the prosecution's evidence was actually independent?

The knife was certainly independent evidence. The prosecution cast some doubt on the defendant's alibi, but that is obviously not probative of guilt.

The only other evidence I can remember is the old man and the old woman's testimony. No doubt what they heard and saw is independent of the other evidence, but I don't think it is fair to consider their identification of the accused to be independent.

Was there other evidence? Or do other people think the identifications are independent?
2.2.2008 12:24am
Malvolio:
IANA statistician either, but it seems wrong to argue that lots of unconvincing evidence somehow adds up to convincing evidence.
A good thing that you aren't a statistician.

Suppose the question is, "Is this coin double-headed?" Flipping the coin and seeing it come up heads would be very unconvincing evidence, I would think. Doing the same thing again would provide just another piece of unconvincing evidence. And another and another.

At some point though, all those unconvincing pieces add up. A hundred heads in a row, and the coin is crooked.

I sat through the flick and I remember thinks, "Kid's guilty." The old lady needs glasses? Well, maybe she's presbyoptic (literally, "old-eyed") and cannot see well close up. So a similar knife could be found? Maybe, but if 1% of all people owned the same kind of knife, that would be the most popular knife in history -- and still the coincidence would be very suspicious. So the old guy got to the door quickly? Maybe he's in good health or was standing closer to the door than previously thought.

I have to agree with the article: the Fonda character was doing what all conspiracy theorists do, trying to conflate several extremely minor possibilities into something resembling a likelihood.

That's why the standard is "reasonable doubt" not "shadow of a doubt".
2.2.2008 12:28am
Elliot Reed (mail):
Aeon and Randy—agreed. I think one of the statistical flaws in Ilya's argument is that he's dramatically overestimating the value of each of those single pieces of evidence. Can it possibly be true that holes in the alibi, or having purchased a similar knife in a store a few days before the murder, makes a defendant 70% likely to be guilty? And as Randy points out, eyewitnesses are extremely unreliable. If we assume, more realistically, that two eyewitnesses separately indicates only a 20% chance of guilt and the other three pieces of evidence separately indicate a 70% chance of guilt and all five pieces of evidence are (implausibly) independent, the chance he's not guilty is 21.95%, which sure looks like reasonable doubt to me.
2.2.2008 12:33am
Elliot Reed (mail):
Oops, I meant that the other pieces of evidence separately indicate a 30% chance of guilt.
2.2.2008 12:34am
Lev:

I really wished I remembered this movie better. How much of the prosecution's evidence was actually independent?



You have to judge that from the jurors' descriptions. This is not a courtroom drama that takes place in a courtroom and a jury room.

It is a jury deliberation drama that takes place in the jury room.


Fonda's character does a good job of showing that no one piece of evidence was enough to prove the accused guilty beyond a reasonable doubt by itself. But he and the other jurors ignore the possibility that guilt might be established by the cumulative weight of multiple pieces of evidence that are individually insufficient.


I think that is an incorrect characterization of what went on. Rather, Fonda's character does a good job of forcing an examination of each piece of evidence that ends up casting doubt as to the piece of evidence's adequacy to show what the prosecution intended it to show. The result was that the cumulative weight of the inadequacies of the multiple pieces of evidence established a reasonable doubt.
2.2.2008 12:38am
Sean O'Hara (mail) (www):
I prefer the Veronica Mars take on the story -- two rich white kids are accused of beating up a prostitute. 11 of the jurors want to acquit, but Veronica feels the defense's case doesn't hold up and she proceeds to argue the rest of the jury to her point of view. Finally there's just one hold-out left, and he says, "You know what, I just want to go home. I'll vote to convict. They'll get off on appeal anyway."

Any lawyerly opinions on "My Cousin Vinny"?
2.2.2008 1:16am
Mikey:
Perhaps I am the only one to infer this, but I've always thought part of the reason the old man and old woman were convinced they heard/saw the defendant was due to their prejudices against the defendant's ethnicity.
2.2.2008 1:45am
Gary Imhoff (mail) (www):
It's completely irrelevant to this discussion, but the best retelling of the Twelve Angry Men plot was Calvin Trillin's story in The New Yorker several years ago about his own jury duty. Trillen writes that he always admired the Henry Fonda character, and tells of how he was able to emulate Fonda when he single-handedly turned around his fellow jury members -- and convinced them not to order in mediocre sandwiches for lunch, but to go to a good Chinese restaurant instead.
2.2.2008 2:11am
Happyshooter:
OJ got off because he was tried before a black jury.

He could have taken the stand, said "Hell yeah I killed 'em, they were white and had it coming" and he still would have been acquited.
2.2.2008 9:24am
Aeon J. Skoble (mail):
Mikey- Yes, that's correct. Yet another reason why their credibility is suspect.
Malvolio - you're implying that if I have a lot of very weak and non-credible evidence before me, I ought to take is as good evidence. That can't be right.
Elliot- this is definitely a higher-level discussion than the last time we talked about this, no? :-)
2.2.2008 9:34am
byomtov (mail):
but this does suggest a common misuse of statistics often used in DNA testing cases. A 99.8% probability for an individual is roughly 2 in 1000. But if there are ten thousand people in the relevant sample, that means that 20 people will have that probability or higher. Which in turn means that the probability that this one individual did it is only 5%, not 99.8%. So this isn't really the right standard.

I see what you are getting at. It is important to explain that the 99.8% figure tells us what percentage of the population is excluded. It is not a measure of the likelihood of guilt. (Though DNA can tell us whether a given individual is excluded or included.)
2.2.2008 9:40am
TaxLawyer:
Prof. Somin,

Your analysis strikes me as incredibly naive, especially as regards the eyewitnesses.

First and foremost, the reliability of eyewitness testimony as a general matter is suspect. Memory is highly fallible and suggestible. Police exacerbate the problem all the time, sometimes innocently (due to incompetence) and sometimes deliberately (due to malevolence).

Second, as to the old man who swears he heard things: an El train passing within feet of your window is not mere "background noise." It seems clear that you've never experienced it (unlike, say, the jurors in the play, several of whom knew that it would have rendered hearing what he claims to have heard impossible).

There's two pieces of evidence out the window entirely (given a reason to discredit a witnessess, I'd suspect the witness is discounted entirely).

The prosecution's case also made much of the "uniqueness" or rarity of the knife. Not so much, aas Henry Fonda showed (and as. I hope, a competent defense attorney would have demostrated in a real trial). Without fingerprints, that makes the probative value of the knife nil.

Whatever evidence is left, it's not enough to convict beyond a reasonable doubt.

I'm afraid your analysis, stripped to its essentials, is nothing more than the all too common belief that the police and prosecutors almost always have it right.

For a view that conforms to yours, however (one of which I assume you're unaware) see Vidmar et al., WAS HE GUILTY AS CHARGED? AN ALTERNATIVE NARRATIVE BASED ON THE CIRCUMSTANTIAL EVIDENCE FROM 12 ANGRY MEN, 82 Chicago-Kent L. Rev. 691 (2007), available at http://lawreview.kentlaw.edu/Contents_82-2.htm)
2.2.2008 11:35am
merevaudevillian:
The pieces of evidence, if I recall (I love watching this film!), were as follows:

1a. The knife, which was called "unique" and "one of a kind" by the storekeeper that sold the boy the knife and found in the father's chest. (Juror #8 found an identical knife while shopping that week.)

1b. The knife was plunged downward into the chest. (It was not shoved upward as could be expected in the average street-fight.)

2. The old woman couldn't sleep, looked across the street, and saw the boy plunge the knife into the father's chest through the last two windows of the elevated train car. (She wore glasses ordinarily but did not have them on at court or, presumably, while in bed.) (Character inferences undercut her testimony.)

3. The old man claimed he heard "I'm going to kill you," a split-second later heard a body hit the floor, rushed to the door within 15 seconds, and saw the boy rushing down the stairs. (The old man walked with a limp and likely would take at least 45 seconds to reach the door.) (Character inferences undercut his testimony.)

4. The boy unable to identify which movies he saw in the theatre (I believe that in the play the explanation was "maybe he slipped in without paying and didn't want to tell the police," and in the movie it was "you wouldn't remember either under a lot of stress.")

Importantly (and I saved this for last), the testimony between (2) and (3) conflicts. The old man could not have heard "I'm going to kill you" if the elevated train were passing at that precise moment. In many ways, Ilya, this tends to present the converse of the situation you described: the individual testimony standing alone might be sufficient, but when considered cumulatively is actually insufficient; someone must be failing to tell the truth.
2.2.2008 11:45am
Dave N (mail):
Any lawyerly opinions on "My Cousin Vinny"?
I like the movie--although I recognize that as comedy, things will not be totally "realistic."

I thought Lane Smith's prosecutor was a dope when he asked a mixed race jury to remember that a word came from Europe along with "all our little ole ancestors." It was a mildly amusing/cringe enducing moment because no southern prosecutor would ever say it.

For a capital homicide, the case goes to trial way too fast, certainly with no time at all for the defense to do any mitigation case.

The "cheat" at the end was unnecessary (the sheriff finding out the real killers had been caught two counties over).

But overall, one of my favorite "legal" movies. Perhaps this legal movie also deserves its own discussion thread.
2.2.2008 12:09pm
Anderson (mail):
Moreover, it's highly unlikely that two separate eyewitnesses would just happenly to mistakenly conclude that they saw (or heard him) if he really wasn't there

I see that TaxLawyer caught the glaring problem here -- Prof. Somin and I have evidently been reading very different reports about eyewitness ID's.

The Simpson case, as ably and angrily recounted by the charmingly narcissistic Bugliosi, is a much better example for the point Prof. Somin is making.
2.2.2008 1:18pm
OrinKerr:
Ilya,

I think you're making two unwarranted assumptions: 1) That the testimony of the eyewitnesses was not influenced by what the government wanted to hear, and 2) That the witnesses at trial were the complete set of relevant witnesses.
2.2.2008 1:27pm
Aeon J. Skoble (mail):
Merevaudvillian-
Thanks for reminding me (and apparently everyone else) about the contradiction between the two witnesses' testimony. That's even more important than the glasses or the limping.
2.2.2008 1:38pm
CheckEnclosed (mail):
Isn't there an attribution issue involved in the Play/Movie? The Fonda and Cobb characters each essentially serve as champions of particular views on guilt or innocence. When the Cobb character's analysis is discredited and his bias is exposed, the jury follows the Fonda character. Isn't this more about small group dynamics than evidence?
2.2.2008 6:22pm
ReaderY:
These numbers assume the pieces of evidence are totally independent of each other, so knowing the result of one piece of evidence tells one nothing about any of the others. But this is very unlikely. There's very likely to be a correlation between some of the pieces of evidence.
2.2.2008 10:20pm
Happyshooter:
I much prefer the Showtime TV recounting of the 12 Angry Men tale, "Hung Jury".

I viewed this short but powerful film while drinking beer late one evening with several fellow lawyers.

The story cuts many of the side plots and reduces the jury to six members. The primary conflict storyline is between the two starring jury members, who are in an initial informal personal relationship but disagree strongly as to the guilt of the accused.

The woman juror, and the rest of the jury, are convinced that the presence of a ring in the shower means the accused is guilty.

Great conflict and pressure results when the lone hold out refuses to vote to convict.

Finally, the lovebird jurors are in his shower and her engagement ring slides off due to the water and their lovemaking. This scene is not just a juror testing the facts and evidence, but also is a final sign the woman juror's planned wedding to an unseen man is off.

Convinced by this experiment, the jury finds the accused not guilty.

The film returns to the courtroom for the joy on the face of the accused as he is released.

Efforts to have CLE credit granted for viewing this movie have been for naught. This is strange, since drinking beer and watching 12 Angry Men and To Kill a Mockingbird gets CLE...
2.3.2008 9:07am
yankev (mail):

The knife, which was called "unique" and "one of a kind" by the storekeeper that sold the boy the knife and found in the father's chest. (Juror #8 found an identical knife while shopping that week.)
I'm trying to remember whether the juror bought it from the very store where the defendant bought his, which would further undercut the storekeeper's testimony.

Of course, with today's courthouse security, the juror would never be able to get the knife past the metal detector -- nor would the defense attorney.
2.3.2008 12:34pm
DAV (mail):

For example, let's assume that the defendant has five pieces of evidence against him and each of them individually shows that there is only a 70% chance of his being guilty. The combined probability of his guilt based on all five items is about 99.8%,


That's only true if you assign a 50% prior probability of guilt. But doing that is tantamount to using the trial itself as evidence against the defendant -- or at least it's making the assumption that the defendant is one of only two possibilities.

The problem is: you don't know how many possible suspects were originally investigated or even if the police had narrowed the investigation prematurely. Suppose the original suspect list was say, 100? Unless you insist on using trial as evidence of guilt, the prior probability should be 1/100. In that case, five pieces of evidence, lending a 70% chance of guilt each, would collectively yield a 41% probability of guilt.

We need to consider that the police and prosecution think they have the preponderance of evidence when bringing the defendant to trial. It's the job of the jury to determine if the prosecution is correct in their belief. If we are to have a presumption of innocence then we need to also remember that the defendant is one of many possible perpetrators and apply that probability as well.
2.3.2008 2:08pm
JeffR (mail):
Wasn't there a case in which a conviction was overturned because the prosecution used a probability argument?

The criminal was driving a red car, the suspect had a red car. If he's innocent, what's the probability that he has a red car? You can estimate that for yourselves, but hypothetically, if the probability is, say, 5%, this is how you'd do the calculations....

If the defendant was innocent, the likelihood is only one in hundreds of thousands that he would fit all the pieces of evidence....

The appeals court pointed out that the prosecution's argument implied that there was probably at least one other person living in the area who fit (not that that was the principal reason that it gave for the decision).

Anyway, DAV is right, and the court was saying, in less mathematical terms, that the prior probability might be one in millions.

It seems like there might be a lot fewer convictions if jurors didn't add one more thing: If he's innocent, what's the probability that he'd be put on trial?

To paraphrase George Stigler, I'm not a lawyer and every day I thank God for it.
2.3.2008 3:36pm