Aarushi Murder Case & Logical Rigor
Last week, on 25th November 2013, Ghaziabad High Court has given the verdict in the Aarushi murder case. The media frenzy in India has not allowed the public interest in the case to die down for the last five and a half year. The verdict as expected was all over the tube. However, I see this as a missed opportunity for media to inform the general public about the process of law to arrive at meaningful conclusions through evidence and logical rigor. The process, I feel, is common in judiciary, science and business analytics. Before I jump to explain my point let me quickly give the gist of the case.
Aarushi Talwar, a 14 year old girl, used to live with her parents Dr. Rajesh and Dr. Nupur Talwar. Both parents are dentists and were at the peak of their careers when Aarushi was murdered in their house on the night of 15th May 2008. She was found lying in a pool of blood with her throat slit. The Talwar family’s 45 year old male domestic help, Hemraj, was the main suspect for the crime. However, the plot thickened when Hemraj was found dead in a similar fashion as Aarushi on the terrace of the Talwars’ house the next day i.e. 17th May 2008. Forensic revealed that both Aarushi and Hemraj were killed around the same time in a similar fashion. Additionally, both parents were present in the house when the crime took place.
The High court has found the Talwar couple guilty of double murder and sentenced them with life imprisonment. People’s opinion about the verdict is completely divided. This is a good case for the media to explain the reason the high court has found the couple guilty. Some people sympathize with the mourning parents, others see them as monsters. This is where media could have explained the case, evidence and the logical progress of the verdict better. However, the media took the easy route and made it into a Bollywood drama. The following are some of the statements I saw repeatedly flashing on the news channels, apparently taken from the verdict.
- Rajesh Talwar was a social drinker and must have drunk on the night of the double murder (so?)
- Parents were not crying when they saw their daughters body (they could be shocked?)
- They did not hug their daughter’s dead body since their clothes were not blood stained (so?)
These seem more like moral judgement rather than the process of law. I am not sure how the high court has arrived at the conclusion but I hope they have firmer arguments for their verdict. Let me also discuss another high profile murder case from the mid-nineties that could be easily termed as the most publicized criminal trial in the American history.
O J Simpson Case and Bayes Statistics
Nicole Brown Simpson and her friend Ronald Goldman were found murdered on Brown’s Bundy Drive condo in the Brentwood area of Los Angeles on 13th June 1994. Nicole’s ex-husband O J Simpson, also an ex-American football player and actor, was the prime suspect for the murders. The trial started with prosecution proving that O J Simpson has a history of physical violence against Nicole. Additionally, they argued that this is a run-up to homicide. To this the defense lawyer, Alan Dershowitz, argued “We knew we could prove, if we had to, that an infinitesimal percentage — certainly fewer than 1 of 2,500 — of men who slap or beat their domestic partners go on to murder them.” Before we analyse this statement by Alan Dershowitz, let me introduce four conditions necessary with their abbreviations (A &B) to understand the problem better.
A : Husband has a history of physical violence against his wife ~A : Husband does not have a history of physical violence against his wife B : Wife got murdered ~B : Wife does not get murdered
If we consider Alan Dershowitz’s argument, the probability or chance of O J Simpson killing Nicole is certainly really low. We are asking chances of wife getting killed if the husband has a history of physical violence against her. This statement can be written probabilistically as
This is conditional probability where we are asking chances of B when A has happened.
However, this is a misinterpretation of facts, a clever trick to fool the non-mathematical audience and the jury. In the end it did work as O J Simpson was acquitted of the charges and set free contrary to the popular belief. The defense lawyer had confused the jury with a wrong question. He asked, how many wife beaters eventually kill their wives? The answer to this question is 0.04%, as mentioned earlier. However, since, the fact available with the court is that the wife is killed; hence the right question to ask over here is ‘how many killed women are murdered by their wife beater husbands?’ This is a completely different question with a completely different answer. To answer this we need to find the conditional probability P (A|B) (Notice this is a flip of P(B|A) that Alan Dershowitz asked). Let us try to answer this question.
According to Alan Dershowitz’s statement 1 in 2500 woman was getting killed by their husband with the history of domestic violence. This is another way of saying that 40 out of every hundred thousand women were getting killed by their husband with history of physical violence against them. Additionally, in the United States in the mid-nineties, the murder rate for women was 1 in 20,000. This means additional 5 in hundred thousand women were getting killed by someone else. Hence the answer to the above question is
This probability is staggeringly higher than 0.04% in the defense lawyer’s argument. We could arrive at the same answer using the following schematic
Nicole Brown is killed and we are asking her chances of getting killed by her husband O J Simpson who also had the history of physical violence against Nicole. Since B has already happened, In this case the two probabilities marked in red are useless to answer the posterior probability P(A|B) . The final Bayesian probability is given as
This probability is the same as the probability we have found using frequency method above. This is an example of logical rigor missing in the O J Simpson murder trial. This error was sighted by mathematicians much after the verdict was announced. I believe that the judicial system across the globe could be improved through transparent analysis of verdicts in public forums. The media has a particularly important role to play in this.
In our last article we have started with a case study from the telecom sector. We learned about cluster analysis using black holes as an analogy. We have not lost the case study in a black hole rather we will continue with the case study in the subsequent articles. See you soon!