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Casey Anthony And The Probability of Guilt

What is the probability that Casey Anthony is guilty of murder? High, but according to twelve men and women honest and true, not high enough. High enough is supposed to be that probability which is “beyond reasonable doubt.” In bringing in a verdict of not guilty, the jurors thus believed there existed reasonable doubt. Just what is that?

This trial swam into my ken about three days ago (during my sojourn in Ithaca, I saw no television; same since then, because I have been traveling). I am no expert on this situation and know few details. I only want to make one point about reasonable doubt, spurred by a comment from Marcia Clark, the prosecutor in the OJ Simpson case, and from one of the jurors in the Anthony trial.

From ABC News:

Casey Anthony juror Jennifer Ford said that she and the other jurors cried and were “sick to our stomachs” after voting to acquit Casey Anthony of charges that she killed her 2-year-old daughter Caylee.

“I did not say she was innocent,” said Ford, who had previously only been identified as juror No. 3. “I just said there was not enough evidence. If you cannot prove what the crime was, you cannot determine what the punishment should be.”…

Ford said that she couldn’t make out “logically” the prosecution’s argument because there were too many unanswered questions about how Caylee died, including how Casey Anthony would have used chloroform to smother her 2-year-old daughter, then put her in the trunk of her car without anyone seeing her.

And now Clark, from the Daily Beast (HT HotAir):

And so, every bit of evidence presented by the prosecution could’ve been tinged with doubt. At the end of the day, the jury might have found that they just couldn’t convict her based on evidence that was reconcilable with an innocent explanation—even if the weight of logic favored the guilty one…

And reasonable doubt? That’s the hardest, most elusive one of all. And I think it’s where even the most fair-minded jurors can get derailed.

How? By confusing reasonable doubt with a reason to doubt. Some believe that thinking was in play in the Simpson case. After the verdict was read in the Simpson case, as the jury was leaving, one of them, I was later told, said: “We think he probably did it. We just didn’t think they proved it beyond a reasonable doubt.” In every case, a defense attorney will do his or her best to give the jury a reason to doubt. “Some other dude did it,” or “some other dude threatened him.” But those reasons don’t necessarily equate with a reasonable doubt. A reason does not equal reasonable. Sometimes, that distinction can get lost.

Listening to Jennifer Ford (at the link above), and noting her (almost certainly loose) use of the word logically, gives weight to Clark’s supposition that the term reasonable doubt has been transmogrified into any doubt.

Now it is a trivial truth that for any contingent event—any real-world event, that is; and murders are real-world events—there can never be logical, 100% certainty that you have proved its cause. That is, it is always possible that your theory of what caused the event can be false. It will always—as in always—be possible to offer alternative theories of what caused the event, no matter how implausible, absurd, or ridiculous these theories are.

In the context of a criminal trial for murder (or any other crime), this means that no theory can ever be proved such that there is no doubt in the theory’s veracity. In other words, no juror, no judge, no person can ever know with 100% certainty that the accused did the deed. They can only know with 99.9999999999999% certainty, or any higher certainty, just never 100% certainty.

This means that if jurors in this case mistook reasonable doubt for any doubt, the defense council best strategy would have been to present other possible theories of Caylee’s death. Which they did. And remember, the absurdity of these theories is besides the point. We just need to know they are possible, no matter how implausible.

Just knowing these other theories exist (and as we saw they are guaranteed to exist) is enough to provide “any” doubt. And that is what appeared to happen here. Juror Ford said as much when she claimed she did now know how Caylee died, that other theories beside murder were possible, even those these other theories were improbable.

In short, Clark apparently was right. Jurors sometimes do misinterpret or misunderstand what reasonable doubt means. Incidentally, reasonable doubt means there exists plausible reasons to believe another theory. I’ll leave it to experts to say whether other plausible theories of Caylee’s death exist.

22 thoughts on “Casey Anthony And The Probability of Guilt Leave a comment

  1. I actually did follow this case rather closely, because it was very similar to a high profile murder case on which I was a juror here in the great state of New Jersey a few years ago. Like the Anthony case, our case was based largely on circumstantial evidence. However, we were able to decide the defendant’s guilt based largely on the testimony of one witness – the medical examiner. He basically asserted that the wounds the victim received could not have been perpetrated by only one person, and since there were only two other people in the room with the victim, there was no plausible theory as to how the victim received the second set of wounds other than that the defendant was guilty. All the rest of the “evidence”, which was weak, didn’t matter.

    I think a lot of these cases boil down to that, and unfortunately for the prosecution in the Anthony case, they had nothing but weak evidence. None of it, on its own, could stand up to the “resonable / plausible doubt” test, which is a fairly high bar considering the way other governments decide innocence or guilt (just look at Italy / Amanda Knox for example). I think the jury, in this case, did come to the right conclusion based on the evidence presented.

    I also think that there is one more factor involved here, and that is the issue of sequestration. The jury had been away from home for over 6 weeks, and IMO, were very much ready to arrive at a speedy decision. I know in the case I was on quite a few jurors were grumbling at deliberating more than one day, and we were not sequestered. I think the fact that juror #3 in the Anthony Case stated that jurors were “sick to their stomachs” and crying over their decision, to me, indicates that they should have taken more time at arriving at their decision. It is important (to me) that a jury is firmly committed to their verdict, and that statement leads me to believe that a good number of them were not in this case.

  2. I am surprised to read the following in your post: What is the probability that Casey Anthony is guilty of murder? High…” It seems to me (and, based on what I’ve read from you in the past, I’d have guessed to you) that the probability that she is guilty is either 0 or 1. What does “probability” mean in such a circumstance. If I say the probability she did it is 0.999 does that mean that, in 1000 randomly selected cases where the known evidence is the same as the Anthony case, in 999 of them the suspect is guilty? This is obviously ridiculous, there will not be a single such identical case and we can never “know” in any event.

    In my company, many of our projects result from an award based on a proposal in response to a request for same. We track these and ask the proposer, prior to award of the project to one of the competitors, to assess the probability that our proposal will be successful. Here again, what is the meaning of “probability”? If I select 75%, does that mean that 75 out of 100 randomly selected proposals with identical characteristics will result in award to my company? There will never be another proposal with identical characteristics.

    I have reconciled this by saying that, if someone assesses the “probability” as 75%, they mean that they are willing to put $3 into a pot where I put $1 and take the $4 if we are successful and give me the $4 if we are not. This interpretation works fine for us, particularly since the purpose of assigning the number is to have a multiplier for the project’s revenue to put into our financial forecasts.

    But this interpretation can’t work in the jury trial situation – we will never know who collects the money. So what in the world does “probability that Anthony is guilty” mean here?

  3. Guy,

    “However, we were able to decide the defendant’s guilt based largely on the testimony of one witness – the medical examiner. He basically asserted that the wounds the victim received could not have been perpetrated by only one person, and since there were only two other people in the room with the victim, there was no plausible theory as to how the victim received the second set of wounds other than that the defendant was guilty.”

    This is hard to understand. Wounds not by one person, Why not two guilty people?

  4. Rob Ryan,

    No. If what you say were true, the probability of any event would be 0 or 1. Probability is a logical measure of certainty, which is always conditional on some evidence, premises, or assumptions. It has nothing to do with repeated events, such as “1000 randomly selected” events, etc.

    When I said “High” I meant that, conditional on my understanding of the evidence, the probability of guilt was near 1. The key is “conditional on my understanding of the evidence”, which I admit is shaky and incomplete. Your knowledge may be different than mine, thus your probability would be different than mine.

  5. Dr. Briggs,

    I appreciate your answer and I’m not simply trying to quibble here – from a epistemological point of view I’m fascinated with this and have been for years. The opportunity to engage someone much more knowledgeable than I is appreciated, thank you.

    On to your reply: I am comfortable with interpretation that the probability with respect to any event that did or did not occur in the past is either 0 or 1, and Anthony’s guilt or lack thereof falls into this category. Certainly (well, almost certainly – I think such as OJ and Jeffrey McDonald did the crimes of which they are accused but have convinced themselves that they didn’t) Anthony can assign 0 or 1.

    Clearly this is not applicable to the situation of our Company’s proposals though – these are future events. So are you saying that in such cases, probability is subjective? If so, how is it to be interpreted? The archetypical “there are 100 marbles in a jar…” scenario can carried out as an experiment and I think that in such a case the probability is objective. But in cases like my Company’s proposals, the assignment of probability seems more analogous to the setting of odds in Las Vegas in a sporting event or the assigning of odds in parimutuel betting at the track.

    It’s interesting also that in, for example, a horse race, the tote board tells the “odds” and then the observation of the race “collapses” the probability of each horse prevailing to 0 or 1. Sounds like quantum mechanics to me.

  6. If I was on the jury I think would have reached the same conclusion as they had. I did follow the case and, although I likely saw things the jury didn’t , the theory that the kid accidentally drowned in the pool and Anthony and her father tried to hide the body was plausible — quite probable even. None of the evidence contradicted that in my opinion. The prosecution also didn’t offer any explanation of why Anthony would have wanted to kill her daughter. Was she a good mother? Not really but that doesn’t mean she’s a murder.

    Besides, if Nancy Grace thinks one way, your best option is to go the other way. 😉

    Juror Ford’s words don’t make much sense if you think about it. She was convinced of Anthony’s guilt but couldn’t bring herself to say so when it mattered? If most juries consist of people like her, how is it that the murder conviction rate is so high? As Briggs mentioned, a reason to doubt is ALWAYS raised.

  7. As a former cop served on a criminal jury once and during deliberations kept quiet for a long time as the rest of the panel struggled with “reasonable” doubt. Most thought it meant “any” doubt. After judge re-read the prepared instructions on that issue about seven jurors “got it”, but four still balked. Took the third reading [of the same material] before they could understand. Suspect the Anthony jury needed the same thing but didn’t get it.

  8. Rob Ryan,

    Think of it this way: if you hold a lottery ticket of which the results have been published and I know the results but you don’t, what probability of winning should you assign to your ticket? Is it any different than it was before the lottery results were published?

    Likewise, the probability that a murder has occurred is 0 or 1. Usually, that is — note that in the Anthony case, you couldn’t even say there WAS a murder. So a probability of murder had to be assigned as well as probability of guilt.

  9. Oh, there were two trials for the two perpetrators. I was on the jury of one of the two trials.

  10. Rob Ryan,

    What is the probability that your proposal will win, and what does that mean.

    You make a model based on your previous experience. Based on the assumption that your model is valid, and the information that you do have, what is the probality that you win. Now, your model may not be particularly quantitative — in which case it would be better to call it a ‘hueristic’ rather than a model. It attemps to quantify the uncertanty of the situation.

    Something perhaps more intuitive, go to your bookie and ask what the odd are on the game. There may be a whole lot of variables that the bookie has considered to determine his line: the players, their previous performances, injuries and the weather. The betting line does not say that if these two teams face each other 100 times, how many games will each team win. It does say how one expert, or a group evaluates the uncertanty of the event.

  11. Rob Ryan,

    Here is my take on your question.

    Based on the epistemological foundations of the classical hypothesis testing, it’s assumed that there is ONE and only one model/process that generates the data/evidence. We therefore can’t assign a number (other than 0 or 1) to the probability P( model is true| evidence ) since the epidemiological foundations implies that either the model is true (she is guilty), or not.

    So instead classical statistics is limited to consider the probability P(evidence | the model is true) (p-value).

    The above is not about the interpretation of the probability though. Given the evidence presented, the juries must have somehow access their degree of belief in her guiltiness.

    My 2 cents.

  12. Rob Ryan,

    JH misleads you. The p-value is not Pr(evidence | the model is true). It is instead Pr (larger statistic than one we saw given infinite repetitions of trial, each randomly different | model true and its parameters take exactly specified values).

    Go back to the Ithaca Teaching Journal posts from about a week ago, and start near the beginning where I define what probability is.

    DAV’s example is also excellent.

    Also, probability does not have to take an exact numerical value. Example, given that “More than half of all Australians wear hats and John is an Australian”, the probability that “John wears a hat” is greater than 50%, and that is the best we can say.

  13. Mr. Rob Ryan,

    I apologize if I misled you.

    ———–
    Mr. Briggs, Mr. Briggs, my Dearest Mr. Briggs,

    I assumed that Mr. Rob Ryan knew what a p-value is. I didn’t mean to define exactly what a p-value is, nor did I intend to interpret it!

    I wanted to provide, perhaps, helpful explanations on “…either she is guilty, or not… 0 or 1”

    Since I feel that you are a dear old pal, like a longtime colleague or a younger brother, so I shall be the normal me and feel free to say the following to you here or in front of you.

    “王九蛋!” (Good luck with Google Translator.)

    Ah… I feel much better already.

    And I am still a fan of yours if you don’t ban me. ^_^

  14. Oh, by the way, Mr. Briggs,

    “Pr (larger statistic than one we saw given infinite repetitions of trial, each randomly different | model true and its parameters take exactly specified values).”

    The above definition is not exactly correct. Ha.. now you may swear at me.

  15. I think that looking at this statistically is kind of misleading. The real question is one of conscience. I almost think “good faith” may be a better phrasing than “reasonable doubt”, as the history of reasonable doubt comes from the religious aspect: If I sentence this person wrongly, I may be punished by my deity, or one’s conscience if nothing else. It would seem to be the better phrasing of the standard anyway, since pedantic people like me get lost in the definition of “reasonable”, but would have a harder time playing with a good faith one.

  16. The instructions in any trial I’ve been a part of defining “reasonable doubt” specifically point out that “reasonable doubt” is not the same as “any doubt”. You have to be a complete brick not to understand that, since it’s read to you, and, generally, read several times in the jury room (I’ve been a juror 6 times in my life already–lucky or unlucky?).

    Reasonable doubt is often defined as doubt for which you can give a reason to the other jurors in the jury room, or doubt for which there is a reason that is articulable (though this standard is sometimes seen as less stringent). NO juror who is paying attention should ever believe that reasonable=any.

  17. Good points. In statistical terms, the question is “where do we set the ‘threshold parameter’ for guilt?” No matter where we set the threshold, there will always be false positives and false negatives, so I think setting it high (“beyond a reasonable doubt”) is best for society. See more about statistics and the Casey Anthony case at
    http://blogs.sas.com/iml/index.php?/archives/167-Statistics-and-the-Casey-Anthony-Case-False-Positives-and-False-Negatives.html

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