Welcome to the second part of this week, where we will study real cases, and focus on concept but on necessary to know how forensic scientist should speak to the court. The aim of this video is for you to understand what the transpose conditional, also called prosecutor fallacy or inversion fallacy is. To be familiar with cases where this error of logic led to appeal, and may have contributed to a miscarriage of justice, and to revisit the need to consider the results given to proposition. Now, we'll leave aside the matter of numerical values and their justification, and we'll focus on the misinterpretation or the statistical value. We will like to present cases involving DNA evidence, where expert, prosecutor, or judges, made serious error of logic in dealing with probabilistic statement. The error we will talk about is known as the prosecutor fallacy, or the transpose conditional, or inversion fallacy. It's a matter of confusion between conditional probabilities. Or just for those wonder what we mean here by conditional probabilities, let us say that it just means that probabilities depend on what we know, what we are told and what we assume. So, if we are told that we need to consider the probability of the result given defense proposition and the information, then this is said to be a conditional probability. The condition is, what is beyond the world given. So here, our condition is given defense proposition and information. The problem we will tackle here is how to present DNA statistics. When DNA profile obtained from a bloodstain is found to correspond to a profile of a given individual, it's customary to provide in a statement the probability for that correspondence to occur by chance. The profile may be found with a probability of say one in two million in the population of interest. This value is commonly refered to as the random match probability. But, when the statistical values is presented, some people erroneously believe that it implied that there is one into million chance that the defendant has not left the bloodstain. You may remember having seen this with Tasha in a previous video. So, it is fallacious to say that there is one into million chance that the defendant has not left the bloodstain. Why? Well, this can be simply explained considering a large enough population, in which one would expect to find several other people with the same profile. For example, among the whole population of Switzerland, about eight million individuals, one would expect on average, to find four persons with the same DNA profile. But, that does not mean that there is a chance of just one in four that the defendant is the source. Note that this error does not occur only in DNA evidence cases. We also encounter it with finger marks, tool marks, fibers, or any kind of forensic results. In fact, such a pitfall of intuition can occur whenever a probabilistic statement regarding the evidence is presented. It is so common that it has been given different names. For example, the prosecutor's fallacy. We need to be careful with this fallacy, otherwise we fail our obligation of logic as we constantly refer to in the end ENFSI guideline. This fallacy has been identified in a few cases, and has led to successful appeals. The cases from England and Wales such as Deen, or Doheny and Adams, or Adams alone, are well-described in the scientific and legal literature. We will concentrate here on the case against Deen, and the case against Adams. But before going into the specifics of these cases, let's have Tasha explain the intricacies of a prosecutors fallacy. Yes, as highlighted by Franco when we discuss the principles of interpretation of forensic science together, you will remember that we said that scientist must give the probability of the results given the prepositions and not the reverse. So, scientists must not give the probability of the prepositions, that is the sole duty of the court. Confusing both probabilities is a fallacy as mentioned by Franco, and it has been coined the prosecutor's fallacy, the inversion fallacy, because we inverse the condition and the event, or the transpose conditional. As everyone is prone to make this error, we will call it the transposed conditional following the step of our mentor, Ian Evett. As this is difficult but crucial topic, the best is to practice together. For example, what do you think about the following expert statement? There is one percent chance that the defendant would have the crime blood characteristics if he was not the source of the stain. Thus, there is a one percent chance that he is not the source. This argument is very appealing, but is it right? Perhaps it's easier to grasp the issue if we modify the example a little. Here is an elephant. What would you answer if I asked you, what is the probability that the animal has four legs given it is an elephant? Would you tell me that is it is very close to 100 percent? I will suppose you do. If not, post something on the forum, and I will say that this probability is assigned a value of 99 percent. Now, let me show you another picture, and ask you another question. What is the probability to be an elephant, if the animal has four legs? You would certainly not say that this probability is 99 percent, right? Most four-legged animals are not elephants, we have cows or which ones, we have dogs, we have wolves, we have giraffes, and the sheeps on our union campus are certainly not elephants, although they have four legs. We see that the answers are different, because the questions are different as well. The difference can be visualize if we use some notation. Sorry, that's what we do as scientists. We have this statement. The animal has four legs, and we are interested in the probability of this statement. We use the letters Pr to denote probability, and put the event we are interested in into paranthesis. This writes as the probability that the animal has four legs. I do not close the parenthesis yet, because all probabilities are conditional. You know that probabilities depend on what we know, what we're told, and what we assume. Here, we are told it is an elephant, so it is given that it is an elephant. Now, in the notation used with probabilities to indicate this given, to indicate this condition, we use a vertical bar like this. Then, we write the condition. So here, we write the probability that the animal has four legs given it is an elephant. Now, you see that I still have not closed my parenthesis. This is because of the first principle of interpretation. As our probabilities depend on our information I need to put this information as well. Indeed, if I had told you that this elephant is Mosha, and that she lost her limb stepping on a landmine in 2006, then you might have had a different probability, right? Because she only has three legs. So, I write the information as well. Here, I write that this animal is not Mosha, it's just a normal elephant. To be efficient I just note the information using the letter I. So, we can read the probability the animal has four legs, given it is an elephant and given the information we have. The comma replaces the word and. I know, it's only two signs less but time is precious. Now, let me write the other statement, which is the probability we were also interested in. What is the probability that the animal is an elephant, given it has four legs and the information. I use the same notation as before and write: the probability the animal is an elephant given it has four legs and the information. Let us look at what we have written. What do we see? In the first question the event elephant is on the right side of what we call the conditional bar. In the second question, the elephant is on the left side. This means that in the first case elephant conditions the probability. In the second case it is the event, four legs that conditions this expression. So, if on the basis of the first statement we say that the second statement is equivalent to the first, then we transpose the conditional. We inverse it. Thus the names inversion fallacy and transposed conditional. At Court, prosecutors also make this error. So, when it is the case we can say it is a Prosecutor's fallacy. So now, let us see the conclusion of our DNA expert. We translate it into probability expressions using the same notation that we saw together. We use the letter E to denote the evidence, and we use HD to denote the defense hypothesis. Just as a reminder, the defense proposition or hypothesis is that someone other than the suspect is the source of the stain, and we also put in the information I. That is, the information that this unknown person is not related to the suspect. So, we have written this down. We have the probability of the evidence given defense proposition and given the information. The expert has given a value to the probability of the evidence given the information and the hypothesis from the defense, in this case the value of one percent. That is the probability of the evidence if the suspect is not the source, so if it is someone unrelated, someone else we do not know. So we have the probability of E given HD and I, which equals one percent. But we cannot say from this value that the probability of HD given E and given I is also one percent. We cannot say that the probability that the suspect is not the source, given the evidence and the information, is one percent. Indeed, imagine that the person was in prison when the alleged facts took place. Then, we are certain it was not him. So, the probability cannot be one percent it is zero. Just as for the animal which had four legs. Even if we know that the probability of an elephant having four legs is 99%, we cannot extrapolate that the probability of being an elephant if the animal has four legs is 99%. This is not logical and it is not sound reasoning. So the transposed conditional occurs when the scientist or the judge or anyone equates the probability of the evidence given HD, or given the hypothesis from the defense, with the probability of HD given the evidence. There are tricks to avoid this fallacy. One is to write things down using the notation we've seen together. Another, is to use what is called in our field Stella's trick. It comes from Stella McCrossan, that we've seen together also and who is an author of a milestone paper on the principle of interpretation. There's also Ian's trick. Ian Evett, who was also part of this publication of course. You will find these tricks in the famous book of John Buckleton, who is also one of my mentors. On the book he wrote on the interpretation of DNA evidence, you will find some of these tricks with the exercises on the platform. Well, I hope that you know now what is a transposed conditional, or what is the prosecutor's fallacy, and I will now pass it over to Franco. Thank you Tasha. We can now come back to the case against Deen, and explore how it occur in practice. For example, in the Deen case in 1993, the court allowed the defendant's appeal because an expert has transposed the conditional by equating one in 700,000 probability of the defendant's DNA match the crime scene DNA. If he is not the source we'd have one in 700,000 probability of the defendant not being the source. The court ruled that the expert should express the evidence in terms of probability of the results and not the probability of the preposition.