Welcome back to the module on applications of probabilities. As I promised you in the last lecture, we're now switching gears towards applications for in the courtroom. A famous, famous problem in legal issues is the so-called prosecutor's fallacies. At the very basic, it's a mistake in logical thinking. In particular in our language in probability, it's the confusion between the probability of A given B versus the probability of B given A. Essentially the question is what has happened already, and what's still uncertain. Remember probability of A given B means condition B holds, what's the probability of A happening? A is still uncertain. B has happened. While in the other way around, probability of B given A, we know A has happened. A is known. But B is still out there, is uncertain. What's the probability of B happening. And this difference can lead to terrible misjudgments. There's a whole legal history now on issues or on cases where this got screwed up. In particular, it's important in the evaluation of evidence. For example in DNA comparisons among possible culprits in the crime. First, I want to explain to you the problem in sort of our language of probability and condition probabilities. And then we're going to look at the example from legal history. Here is the question. We have a defendant in court, is he or she innocent? So the event I says the defendant is innocent, the opposite I complement is guilty. So let's just look at innocent versus guilt, nothing in between. So don't tell me I smoked, but didn't inhale. No, that doesn't exist here. Innocent versus not innocent, I complement. And then there's an issue of what I call the DE, the damning evidence. The really bad evidence that we have, and why we are in court. This could, for example, be the DNA match at the crime scene. Now let's look in our language, what do the two different condition probabilities mean? First, the probability of the damning evidence given the person is innocent. So that's a probability that an innocent person, so that's what we assume, we know person is innocent, matches the damning evidence. Now whether he or she matches the damning evidence, that's now the uncertain part. That's where we still have a probability. Now let's think about it the other way around, what the probability of I given DE? The probability of being innocent given the damning evidence. That now means we have the person who has the damning evidence. Maybe his or her DNA, matches something found at the crime scene. Also DE has happened and now the question is, is this person innocent or guilty? So that's still out there. Notice the difference between those two probabilities. Here's now the heart of the problem. You go to court, a prosecution's expert witness says, an innocent person has a 1 in 100,000 chance of matching the damning evidence. And now comes a prosecutor's fallacy. Now the prosecutor says the defendant has the damning evidence. Therefore, there's only a 1 in 100,000 chance that the defendant is innocent. Clearly he's guilty. Jury, judge, I could say the defendant is guilty. Send him to prison or her to prison. That's a prosecutors fallacy and the thinking is wrong. Let's look at what the expert witness said. The expert witness said the probability of an innocent person, that's if the person is innocent, has the damning evidence is 1 in 100,000. The prosecutor's fallacy is to turn this around and say, the damning evidence match is there, that's why now the probability of innocence is also 1 in 100,000. But that's wrong. Because in that argument the prosecutor changed the uncertain statement that's still uncertain and the condition around. But those two probabilities are usually different. Now you say that's a problem. How do we get the other probability? We have Bayes' rule. You remember Bayes' rule from this flipping probabilities of the previous module? We actually have at least a theoretical way to go from the expert witnesses statement to the probability that the prosecutors, the jury, and the judge care about. Remember the flipping formula. So probability of I given damning evidence equals the probability of damning evidence given I times these two other probabilities or this ratio of two other probabilities. Now the question is what is the probability of the left hand side? That's really the one we have found damning evidence against someone but what's the probability that this person could still be innocent or the probability that this person is guilty? On the right hand side, we have the first probability. That's what the expert witness said, that's a tiny, tiny probability. But wait a minute! This now gets multiplied by a ratio. If that probability of DE is very small, we're dividing by a number that's very small. And a small number divided by another small number can suddenly get a little larger. Let me illustrate this in a simple example. The guilty person is among 500,000 people living in a town. Let's say we know this, well, because of circumstantial evidence or timing, someone was robbed or murdered or something, we know. We are down to 500,000 people and let's make the following assumption. The guilty person left some DNA or some match, that boot that guilty person also has and that's our damming evidence. So here now probabilities we know. And so simplification, let's assume there's a single culprit. There is only one criminal. We don't have a group of criminals that's a different sum altogether. So there is a single person that's guilty and the rest is innocent. So that gives me the probability of being innocent and the probability of guilty. As you see here, 499,999 out of 500,000, those are all innocent and that last person is the guilty one. And we know if you're the guilty person, if Ic is true, the complement of innocent Ic, the guilt, then the probability of showing the damning evidence is one. Using these probabilities, I can now use base rule. I can calculate the probability of the damming evidence. And I can calculate the probability of being innocent, given the damning evidence and being guilty, given the damning evidence and you have the probabilities here. So, there's a one in six chance that given the damning evidence has been found, the person is guilty, and there's a five in six chance that the person is innocent. The calculations are here, you can look at them. So look at this. There's still, we find the damning evidence for a person. The probability that he or she is still innocent is five in six. What's the logic here, how can that be? Here's another way of looking at it. Given the probabilities that we have, so we have 500,000 people and the probability is of any innocent person having this damning evidence, here at the first calculation is 1 in 100,000. So let's do simple math. You have 500,000 people and a 1 in 100,000 chance that an innocent person catches, has that same DNA, or the same sort of circus shoe or something that was found at the crime scene. 500,000 people, 1 in 100,000 chance of someone by accident having this. That means in that city you've five innocent people who match that stuff, plus a guilty person, that's six people who match them. One of those six is a bad guy, the other five are the unlucky guys. So that's now the idea is that because the chance that an innocent person has this damning evidence, even if it's small, if the number is large enough of potential criminals, and therefore also potential innocent people, that's how we get, in the end, quite a large group. In this case, six potential criminals, but only one did it. And so even if you have that wrong DNA, there's still a good chance you are innocent. And that's sort of intuitive explanation here, that for some of you, may be a little easier than just going through the tedious math of applying base rule. And here, so I'll summarize this in probability table, perhaps this is a little more familiar for some of you now that we have worked with probability tables. That's sort of summarizes all the probabilities. Now, you may think this is kind of obscure and never happens. Sorry, no, that's not the case. If you google prosecutors fallacies you will see many, many examples. I picked out one that's sort of a well known in legal history, a famous one. And that's a group, it's the so called Birmingham Six. So, let me tell you a little story. Remember back in the 70s, there were still a lot of or many issues in the United Kingdom between the Protestants and the Catholics in Ireland and Northern Ireland. And there was a terrorist organization called the IRA that organized bombings and killed British people. In particular, a bad, bad crime happened in the second largest city of England, Birmingham, in November of 74. There was a bad pipe bombing where 21 people were killed and 182 were injured. Quickly, six men were found who became infamous as Birmingham Six in the UK press. And they were found guilty of this murder, less than a year later in the August of 1975 and each defendant received 21 life sentences. The key statement, why these people were found guilty is the following. A forensic scientist had said that among these six defendants, they all had to do tests. And he was sure, up to 99% that two out of these six Birmingham six people had handled explosives. So here was his claim in our language. The probabilty of handling explosive given that his test, his special test is positive, is 0.99. And people said, that's so high, clearly now these bad guys handled the explosives and they must be the terrible criminals. Now, it's not 100% its only 99% and because these tests are not correct. In particular, at the time it was known that other substances produce positive test results including nitrocellulose, I don't know whether that I pronounced that correctly. This chemical is present in paint, on playing cards, in gasoline, cigarettes and soap. Now here is a problem of what happened. The expert witness already made the prosecutor fallacy because what he should have said is the following. If you handle explosives, then the test will be positive. So, then you'll come in touch this chemical and the test will give us a positive signal. But the claim that he made, if the test is positive that means you handed explosives, seems to be much smaller than many of us on any given day get in touch with gasoline, if you go to a gas station, for smokers among you, you may get in touch with if you smoke. If it's in soap, well many of us get in touch with that, and other reasons. In particular the defendants had been caught on a train playing cards. So what happened eventually after various appeals, finally in 1999 there was a second appeal and they were all released from prison and actually now days people think that these people were actually innocent and that other people, and apparently there is some known names, but these people didn't get convicted for doing the crime. There were other issues at these various appeals and these various trials but it was very hard with this misstatement from the forensic expert who had already done the flipping that he shouldn't have done off the condition of probabilities, and he committed the prosecutors fallacy. And this is just one famous example of many others in the legal history where people confuse what's given, what have we found, and what still out there. And what we have found is the evidence of the test result, what's unclear is the innocence or guilt but people often then argue with it the other way around. Whenever we have a prosecutor's fallacy, that's a neglected application of base rule. People are forgetting about that they really should use base rule. Next, and as our last application, in the next lecture, I will show you a particularly sad story of a real tragedy where a prosecutors fallacy lead to a lot of harm. So please come back for the last application of this module. Thank you.