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Welcome back to the module on applications of probabilities.

Â As I promised you in the last lecture,

Â we're now switching gears towards applications for in the courtroom.

Â A famous, famous problem in legal issues is the so-called prosecutor's fallacies.

Â At the very basic, it's a mistake in logical thinking.

Â In particular in our language in probability, it's the confusion

Â between the probability of A given B versus the probability of B given A.

Â Essentially the question is what has happened already, and

Â what's still uncertain.

Â Remember probability of A given B means condition B holds,

Â what's the probability of A happening?

Â A is still uncertain.

Â B has happened.

Â While in the other way around, probability of B given A, we know A has happened.

Â A is known.

Â But B is still out there, is uncertain.

Â What's the probability of B happening.

Â And this difference can lead to terrible misjudgments.

Â There's a whole legal history now on issues or

Â on cases where this got screwed up.

Â In particular, it's important in the evaluation of evidence.

Â For example in DNA comparisons among possible culprits in the crime.

Â First, I want to explain to you the problem in sort of our language of

Â probability and condition probabilities.

Â And then we're going to look at the example from legal history.

Â Here is the question.

Â We have a defendant in court, is he or she innocent?

Â So the event I says the defendant is innocent,

Â the opposite I complement is guilty.

Â So let's just look at innocent versus guilt, nothing in between.

Â So don't tell me I smoked, but didn't inhale.

Â No, that doesn't exist here.

Â Innocent versus not innocent, I complement.

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And then there's an issue of what I call the DE, the damning evidence.

Â The really bad evidence that we have, and why we are in court.

Â This could, for example, be the DNA match at the crime scene.

Â 2:27

Now let's look in our language,

Â what do the two different condition probabilities mean?

Â First, the probability of the damning evidence given the person is innocent.

Â So that's a probability that an innocent person, so that's what we assume,

Â we know person is innocent, matches the damning evidence.

Â Now whether he or she matches the damning evidence, that's now the uncertain part.

Â That's where we still have a probability.

Â 2:58

Now let's think about it the other way around,

Â what the probability of I given DE?

Â The probability of being innocent given the damning evidence.

Â That now means we have the person who has the damning evidence.

Â Maybe his or her DNA, matches something found at the crime scene.

Â Also DE has happened and now the question is, is this person innocent or guilty?

Â So that's still out there.

Â Notice the difference between those two probabilities.

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Therefore, there's only a 1 in 100,000 chance that the defendant is innocent.

Â Clearly he's guilty.

Â Jury, judge, I could say the defendant is guilty.

Â Send him to prison or her to prison.

Â That's a prosecutors fallacy and the thinking is wrong.

Â Let's look at what the expert witness said.

Â The expert witness said the probability of an innocent person,

Â that's if the person is innocent, has the damning evidence is 1 in 100,000.

Â The prosecutor's fallacy is to turn this around and

Â say, the damning evidence match is there,

Â that's why now the probability of innocence is also 1 in 100,000.

Â But that's wrong.

Â Because in that argument the prosecutor changed

Â the uncertain statement that's still uncertain and the condition around.

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Now you say that's a problem.

Â How do we get the other probability?

Â We have Bayes' rule.

Â You remember Bayes' rule from this flipping probabilities of

Â the previous module?

Â We actually have at least a theoretical way to go from the expert

Â witnesses statement to the probability that the prosecutors,

Â the jury, and the judge care about.

Â Remember the flipping formula.

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So probability of I given damning evidence equals

Â the probability of damning evidence given I times these two other probabilities or

Â this ratio of two other probabilities.

Â Now the question is what is the probability of the left hand side?

Â That's really the one we have found damning evidence against someone but

Â what's the probability that this person could still be innocent or

Â the probability that this person is guilty?

Â On the right hand side, we have the first probability.

Â That's what the expert witness said, that's a tiny, tiny probability.

Â But wait a minute!

Â This now gets multiplied by a ratio.

Â If that probability of DE is very small,

Â we're dividing by a number that's very small.

Â And a small number divided by another small number

Â can suddenly get a little larger.

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The guilty person is among 500,000 people living in a town.

Â Let's say we know this, well, because of circumstantial evidence or

Â timing, someone was robbed or murdered or something, we know.

Â We are down to 500,000 people and let's make the following assumption.

Â The guilty person left some DNA or some match,

Â that boot that guilty person also has and that's our damming evidence.

Â So here now probabilities we know.

Â And so simplification, let's assume there's a single culprit.

Â There is only one criminal.

Â We don't have a group of criminals that's a different sum altogether.

Â So there is a single person that's guilty and the rest is innocent.

Â So that gives me the probability of being innocent and the probability of guilty.

Â As you see here, 499,999 out of 500,000,

Â those are all innocent and that last person is the guilty one.

Â And we know if you're the guilty person, if Ic is true,

Â the complement of innocent Ic, the guilt,

Â then the probability of showing the damning evidence is one.

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Using these probabilities, I can now use base rule.

Â I can calculate the probability of the damming evidence.

Â And I can calculate the probability of being innocent,

Â given the damning evidence and being guilty,

Â given the damning evidence and you have the probabilities here.

Â So, there's a one in six chance that given the damning evidence has been found,

Â the person is guilty, and there's a five in six chance that the person is innocent.

Â The calculations are here, you can look at them.

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So look at this.

Â There's still, we find the damning evidence for a person.

Â The probability that he or she is still innocent is five in six.

Â What's the logic here, how can that be?

Â Here's another way of looking at it.

Â Given the probabilities that we have,

Â so we have 500,000 people and the probability

Â is of any innocent person having this damning evidence,

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here at the first calculation is 1 in 100,000.

Â So let's do simple math.

Â You have 500,000 people and

Â a 1 in 100,000 chance that an innocent person catches, has that same DNA, or

Â the same sort of circus shoe or something that was found at the crime scene.

Â 500,000 people, 1 in 100,000 chance of someone by accident having this.

Â That means in that city you've five innocent people who match that stuff,

Â plus a guilty person, that's six people who match them.

Â One of those six is a bad guy, the other five are the unlucky guys.

Â So that's now the idea is that because

Â the chance that an innocent person has this damning evidence, even if it's small,

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if the number is large enough of potential criminals,

Â and therefore also potential innocent people, that's how we get,

Â in the end, quite a large group.

Â In this case, six potential criminals, but only one did it.

Â And so even if you have that wrong DNA,

Â there's still a good chance you are innocent.

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And here, so I'll summarize this in probability table,

Â perhaps this is a little more familiar for

Â some of you now that we have worked with probability tables.

Â That's sort of summarizes all the probabilities.

Â Now, you may think this is kind of obscure and never happens.

Â Sorry, no, that's not the case.

Â If you google prosecutors fallacies you will see many, many examples.

Â I picked out one that's sort of a well known in legal history, a famous one.

Â And that's a group, it's the so called Birmingham Six.

Â So, let me tell you a little story.

Â Remember back in the 70s, there were still a lot of or

Â many issues in the United Kingdom between the Protestants and

Â the Catholics in Ireland and Northern Ireland.

Â And there was a terrorist organization

Â called the IRA that organized bombings and killed British people.

Â In particular, a bad, bad crime happened in

Â the second largest city of England, Birmingham, in November of 74.

Â There was a bad pipe bombing where 21 people were killed and 182 were injured.

Â Quickly, six men were found who became infamous

Â as Birmingham Six in the UK press.

Â 11:48

And they were found guilty of this murder,

Â less than a year later in the August of 1975 and

Â each defendant received 21 life sentences.

Â The key statement, why these people were found guilty is the following.

Â A forensic scientist had said that among these

Â six defendants, they all had to do tests.

Â And he was sure, up to 99% that two out of these six

Â Birmingham six people had handled explosives.

Â So here was his claim in our language.

Â The probabilty of handling explosive given that his test,

Â his special test is positive, is 0.99.

Â And people said, that's so high, clearly now these bad guys

Â handled the explosives and they must be the terrible criminals.

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Now, it's not 100% its only 99% and because these tests are not correct.

Â In particular, at the time it was known that other substances produce

Â positive test results including nitrocellulose,

Â I don't know whether that I pronounced that correctly.

Â This chemical is present in paint, on playing cards,

Â in gasoline, cigarettes and soap.

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So, then you'll come in touch this chemical and

Â the test will give us a positive signal.

Â But the claim that he made, if the test is positive that means you handed explosives,

Â seems to be much smaller than many of us on any given day get

Â in touch with gasoline, if you go to a gas station, for smokers among you,

Â you may get in touch with if you smoke.

Â If it's in soap, well many of us get in touch with that, and other reasons.

Â In particular the defendants had been caught on a train playing cards.

Â So what happened eventually after various appeals,

Â finally in 1999 there was a second appeal and they were all released from prison and

Â actually now days people think that these people were actually innocent and

Â that other people, and apparently there is some known names, but

Â these people didn't get convicted for doing the crime.

Â There were other issues at these various appeals and these various trials but

Â it was very hard with this misstatement from the forensic expert who had already

Â done the flipping that he shouldn't have done off the condition of probabilities,

Â and he committed the prosecutors fallacy.

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And this is just one famous example of many others in the legal history

Â where people confuse what's given, what have we found, and what still out there.

Â And what we have found is the evidence of the test result, what's unclear

Â is the innocence or guilt but people often then argue with it the other way around.

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Next, and as our last application, in the next lecture,

Â I will show you a particularly sad story of a real tragedy

Â where a prosecutors fallacy lead to a lot of harm.

Â So please come back for the last application of this module.

Â Thank you.

Â