Reasoning about probability

There are many ways to approach the thorny issue of “reasonable doubt“.

Jury instructions may give guidance in this way:

“The State has the burden of proving the defendant guilty beyond a reasonable doubt. This means the State must prove each element of each charge beyond a reasonable doubt. In civil cases, it is only necessary to prove that a fact is more likely true than not or that its truth is highly probable. In criminal cases such as this, the State’s proof must be more powerful than that. It must be beyond a reasonable doubt. Proof beyond a reasonable doubt is proof that leaves you firmly convinced of the defendant’s guilt. There are very few things in this world that we know with absolute certainty, and in criminal cases the law does not require proof that overcomes every doubt. If, based on your consideration of the evidence, you are firmly convinced that the defendant is guilty of the crime charged, you must find her guilty. If, on the other hand, you think there is a real possibility that she is not guilty, you must give her the benefit of the doubt and find her not guilty.”

Estimates suggest that about 5% of people convicted and sentenced to death are innocent, meaning juries are regularly making mistakes, thinking a case for guilt has been proved beyond a reasonable doubt when this is not the case.

The reasons for these errors are undoubtedly complex and varied, but here I want to concentrate on fallacious reasoning about probability. Sometimes people talk about “too many coincidences”, when I hear that claim I immediately suspect fallacious reasoning.

How should a jury deliberate? Well, it would be good to reduce that error rate from 5% to a more acceptable rate, let’s say 1 in 1,000. That would be 500 times better than the current rate, and a significant improvement.

So let’s say we will convict if the chance of innocence is less than 0.1%, recognising that we cannot overcome every doubt. To put it another way, the probabilty of guilt should be more than 0.999.

But how can we estimate this probability in a rational way?

One thing to beware of is something known as the “prosecutors fallacy”.


At its heart the fallacy involves assuming that the prior probability of a random match is equal to the probability that the defendant is innocent. For instance, if a perpetrator is known to have the same blood type as a defendant and 10% of the population share that blood type, then to argue on that basis alone that the probability of the defendant being guilty is 90% makes the prosecutor’s fallacy (in a very simple form).

The basic fallacy results from misunderstanding conditional probability and neglecting the prior odds of a defendant being guilty before that evidence was introduced.

In other words, we must look at the “prior odds” of a defendant being guilty.

Let’s take a concrete example, the case of Jodi Arias. This may have looked like a simple case to a casual observer. Travis Alexander’s body was found dead in the shower at his home in June 2008. He had been shot in the head and stabbed repeatedly. Photographs recovered from a camera that was found in the washing machine showed Arias had visited Alexander on 4 June 2008, and pictures taken by the same camera showed Travis in the shower, around 5:30pm and then just a minute later, lying on the flood bleeding. Moreoever blood recovered from the scene contained a mixture of Arias’ DNA and Alexander’s DNA. However Arias when questioned first denied having visited Travis, then when shown the photographs claimed that intruders had appeared and shot and killed Alexander.

Finally she claimed that she had acted in self defence, that Alexander had become angry after she dropped his camera, and he had been shot after he assaulted her, then chased her, and she pointed a gun at him to keep him away ( explaining that she had squeezed the trigger without meaning to ). After he was shot, she claimed to be unable to remember very much about what happened, other than an overwhelming sense of fear.

The prosecution asserted that Arias’ account was untrue, and that Arias had travelled to Mesa with the intention of killing Alexander, evidenced by measures she took to keep the visit secret.

At first sight the defense case seems highly improbable. Why would Arias not have immediately told the police what happened if she acted in self defense? Why would Alexander become violent simply because she dropped his camera? And how if she was defending herself, could she have stabbed Alexander so many times, ultimately slashing his throat? And how could she have almost no memory of what happened after Alexander was shot?

Here there is a danger of what I would call a “lack of imagination”. The universe is full of events that are hard to comprehend, which may seem unusual, paradoxical and unlikely. For example, how did life start? Life is very complex, and it is hard to comprehend how such complex assemblies of atoms could come together by chance.

But we should also consider the prosecution version of events. The prosecution claimed that Arias had concealed her trip to Mesa, and there was circumstantial evidence to support this hypothesis. Travis was shot by a 0.25 caliber bullet, and  a 0.25 caliber gun had been stolen from Arias’ home seven days earlier. Secondly Arias had borrowed two gas cans from a friend, and bought a third gas can, and there was no record of her buying gas while in Arizona. Thirdly, Arias had turned off her mobile phone shortly before entering Arizona, perhaps in an effort to conceal the visit. Finally, when Arias was stopped by a policeman in Salt Lake City the day after Travis was killed, one of her licence plates was upside down, and the other was missing ( it was inside her car ). Could all this evidence of premeditated murder simply be a series of coincidences?

A reasonable person might easily conclude that this is overwhelming evidence of guilt. Firstly Arias’ denials, and secondly circumstances suggesting that she took measures to keep her visit to Mesa secret.

There is one big, massive problem here though. It can be expressed as a single word : why? Why would Arias want Alexander dead, and be prepared to risk ruining her life by being convicted of his murder?

In other words, the “prior odds” mentioned above are massively AGAINST Arias being guilty.

So, now to the question of how we could estimate the probability of guilt in the case, based only on the evidence given above.

There is no real doubt that Arias did kill Alexander. Firstly she testified that she killed him. Secondly her story that intruders appeared at the moment when Alexander was shot is not credible.

But is it possible that she was first attacked by Alexander? If there is a fight between two people, it’s reasonable to suggest that either could have started the fight, from that perspective, the chance that Arias is guilty is only 0.5.

The big issue here though is the improbability, before the evidence is considered, that Arias planned to murder Alexander. Planned murder is a very serious criminal offence, and very rare as well. Looking at women on death row in the United States, or the history of crime, there are almost no cases that could be said to be similar examples, where a woman has planned many days in advance to murder a man for no credible reason.

There was apparently no financial motive – although Arias did owe Alexander some money, it was not a large sum, and few people have even suggested that as a motive. Instead, the suggestion is that Arias may have acted due to feelings of jealousy or rejection. It is not unusual for men to murder women for this reason, especially at the end of a relationship. However cases of women murdering men for this reason seem much more unusual, and there seem to be no precedents for a woman who was not even living with the man at the time, and who had no reason to feel unreasonably rejected, to have embarked on murder.

From this perspective, the likelihood that Arias is guilty of murder planned a week or more ahead is very, very low, perhaps 1 in a million, or even 1 in a billion, perhaps even less, depending on your point of view.

So, the question then becomes, do the apparently improbable suspicious circumstances suggesting Arias concealed her trip to Mesa, combined with the apparent improbability of the defense case,  mean that it is very probable this is premeditated murder?

Let’s be generous to the prosecution here. Firstly, let’s say that the chance of such a suspicious event such as the theft of a gun of the correct caliber from Arias’s home a week before the murder is quite small. Perhaps 1 in 100. There is a problem here, in that the universe of possible “suspicious events” is very large. What if the theft had been two weeks before? That would presumably still be suspicious. What if the theft had been from the house next door, would that be suspicious? This consideration must tend to reduce the weight of this circumstance.

Now what about other circumstances? Well, I suggest that the other circumstances are really not at all conclusive. It is not unusual to switch a mobile phone off to save the battery. Paying for gas with cash is perhaps unusual ( which is what Arias claimed ). However having irregular license plates seems more likely to attract attention than be part of a plan of concealment, and the explanation Arias gave that pranksters were responsible seems more plausible to me.

So let’s take stock. Taking a view favorable to the defence:

Arias’ final story is in fact quite possible, with some imagination.

Likelihood that Arias planned the murder a week in advance for some very unusual reason : 1 in a billion.

Likelihood that the gun theft and other circumstances are just coincidences : 1 in a thousand.

Likelihood that Arias is guilty : only 1 in a thousand. Not nearly enough for a conviction.

Of course this process has been massively subjective. It is a matter of opinion how improbable the circumstances are, how improbable the defense case is, or how improbable it is that Arias is capable of premeditated murder planned a week in advance.

This is not the end of the matter though. There is a lot of evidence that has not yet been taken into account. The purpose of this article was to show the danger of believing the “too many coincidences” argument. We must not consider only circumstances that make the prosecution case seem likely, the general improbability of the prosecution case must also be taken into account.

I believe this is where juries mostly go wrong. There are numerous examples of defendants being convicted in spite of having a solid alibi. Many juries seem blind to exculpatory evidence or circumstances, and perhaps do not understand the idea of prior odds.

[ Please see here for a much more complete description of the evidence in this case ]

See also How juries are fooled by statistics a TED talk by Peter Donnelly.

11 thoughts on “Reasoning about probability”

    1. Opportunity – Juan said that during the trial Jodi had opportunity to shoot Travis with a gun. That fact that Jodi had opportunity should be enough to convict Jodi of murder. But lets look at the opportunity. Did opportunity exist for Jodi to have shoot Travis with the gun stolen from her grandpa’s house? Wouldn’t be possible because the gun had been stolen prior to the murder of Travis. How can Juan prove opportunity with the stolen gun when Juan did not prove in court before the jury that Jodi had in fact stolen the gun from her grandpa’s and later purchased round nose bullets to shoot Travis with? In order for this opportunity to have be proven Juan needed to prove not only that Jodi had stolen the gun but had also purchased round nose ammo to replace the hollow points with that had been in the gun stolen from her grandpa’s house. If Juan can prove that Jodi stole the gun from her grandpa’s house and round nose ammo then he can more than likely prove that the gun used to shoot Travis with had been the one stolen from Jodi’s grandpa’s house. But since Juan did not prove either of these issues in court then how can Juan prove that Jodi had opportunity based on a stolen gun that did not exist at the time of the murder of Travis? How can you prove opportunity with something that can’t be proven to exist? Juan’s theory of Jodi having opportunity based off of the stole gun is speculation. Speculation is not truth unless tests are conducted to prove that the speculation is in fact real based on the evidence passing numerous scientifically based tests in a lab setting. Scientifically meaning science and not ” A word that can mean anything. ” type of testing.

      Juan can’t prove opportunity based off of Travis having a gun either for the simple fact that Juan did not prove that Travis owned a gun during the trial to say that Jodi would have opportunity because Travis had owned a gun that he and Jodi both knew about. Granted Jodi said that the she had seen a gun owned by Travis in Travis’ house in one of her journals but unless Juan can prove that Travis did in fact own a gun either legally or illegally then Juan did not prove opportunity once again. Like the stolen gun how can Juan prove opportunity with a gun that wasn’t proven to have existed at the time of the murder of Travis?

      In order to prove opportunity Juan had to prove that either gun existed within the same day that the murder take place. If neither gun was proven to have existed at the time of the murder then there was never any opportunity.

      1. Jodi testified that she shot Travis with a gun he kept on a shelf. However the prosecution disputed this. I agree the prosecution failed to prove beyond a reasonable doubt that the gun that shot Travis was the gun stolen in Yreka. Depending on whether you adopt a strict approach to proof or a more probablistic approach ( either can be justified ), the gun stolen in Yreka could be excluded from consideration. At most Juan showed the possibility that the stolen gun was the one that shot Travis. It then comes down to questions of “improbable coincidences”, in other words probability.

  1. I don’t know why you find the intruder story not credible. If there were others who had a motive to kill the guy, it’s at least plausible.

  2. I don’t see anything ruling out intruder(s), or even calling the idea into question. The “mixed” DNA means nothing if she was there at the time, which is not inconsistent with an intruder scenario.

    The linked article doesn’t seem to add up to anything on that score either, but maybe I’m missing something.

    1. The evidence shows Travis standing at the sink after being shot. The only reasonable explanation is the shooting was (to some extent) unintentional, an accident.

      You seem to be ignoring the subject of the article, which is “prior odds”.

  3. I suppose I am. I don’t have any problem with the idea. I’m in complete agreement that she seems like a highly unlikely murderess. But as I’ve said, I don’t put much stock in her claims that she killed TA. Aside from that I was just wondering why you would find the intruder idea not credible, but I guess we just disagree about that.

    1. Well, I consider Arias’ (eventual) confession to be a very credible and plausible explanation of Travis’ death, and one that fits the evidence recovered from the scene perfectly.

      There is a lot of confimation of her account, in terms of blood evidence, DNA evidence, her having a cut finger from using a knife, the photo of the shower ceiling, Travis having an anger/resentment issue towards her (for example he suspected she had slashed his tires and had spied on him, and he was abusive in text messages).

      Whereas intruders suddenly appearing and murdering Travis for a reason we cannot even guess at is not at all likely.

  4. Mike LeBlanc asks:

    “So you believe she cut her finger there in the bathroom on June 4th 2008?”

    Yes, I believe Jodi cut her finger using the knife on June 4, 2008, that’s a very common injury when stabbing with a knife with no hilt. There was some complication, another finger was cut previously in an accident at a restuarant where she worked.

    There were traces of blood found downstairs, likely from this cut continuing to bleed.

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