Majority in Halacha and in General 2, Lesson 9
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Table of Contents
- Biases, slander, and the distinction between fact and Jewish law
- The presumption of innocence, “the burden of proof rests on the claimant,” and why this is not statistics
- Venn diagrams and the representativeness fallacy as a formal description
- Rava, Rava’s wife, trust in judges, and Maimonides’ approach
- “The majority has been undermined” in Yevamot and Ketubot as conflicting majorities and a subset
- The PISA example and the claim that almost every news number falls into this fallacy
- A rare disease, a 99% test, and a fishing net of the wrong size
- The Roy Meadow case, crib death, and Munchausen syndrome by proxy
- 99% evidence in criminal law, confession, and “some supporting element”
- Expected value, the St. Petersburg paradox, and the probabilistic critique of Pascal’s wager
Summary
General Overview
The text presents the representativeness fallacy and base-rate biases as statistical-judicial distortions in which one applies the distribution of a general population to a case that belongs to a subgroup with additional information, so the relevant distribution changes and sometimes even reverses. It connects this to the distinction between the factual plane and the halakhic / legal plane, and argues that many presumptions and rules of evidence are not really based on probability at all, but on institutional considerations of managing a legal system, deterrence, and creating rules of the game. It illustrates this fallacy in monetary law, in the presumption of innocence, in Talmudic passages where “the majority has been undermined,” in medical testing for rare diseases, and in statistical conviction in the crib-death case in England. Finally, it argues that even when an expected-value calculation is correct, as in the St. Petersburg paradox and Pascal’s wager, making decisions by expected value alone may still be mistaken when that expectation relies on rare tail events.
Biases, Slander, and the Distinction Between Fact and Jewish Law
The text opens with biases that Daniel Kahneman calls representativeness fallacies and base-rate biases, and explains that the principle is the mistake of applying general statistics to a case about which there is classified or filtering information. It brings the laws of slander—“you may not believe it, but you should be concerned about it”—as an example of the distinction between a factual assessment and a halakhic prohibition against accepting a claim as knowledge, and explains that on the factual plane there may be a high chance that the rumor is true, and yet on the halakhic plane one is still forbidden to “believe” it. It presents the difference between a judicial institution and an investigative institution, and argues that a single witness does not go to a religious court in order to punish, but may go to the police in order to locate an additional witness and prevent acts from continuing. It notes that in Israeli law, “fruit of the poisonous tree” is not, in principle, disqualified as evidence, and with respect to Jewish law he says he is not aware of such a disqualification as a matter of basic law, and that if it is evidence, then it is evidence, and only afterward does one discuss the transgression involved in obtaining it.
The Presumption of Innocence, “The Burden of Proof Rests on the Claimant,” and Why This Is Not Statistics
The text uses the debate surrounding the indictment against Bibi to sharpen the point that the presumption of innocence on the legal plane is not identical to the question of the factual probability of guilt, and that in the public sphere one may “be concerned” even without determining criminal guilt. It presents a common explanation of the presumption of innocence as resting on the fact that most people are innocent, and then argues that this is a statistical fallacy, because the defendant belongs to a subgroup that has already been filtered through procedures and evidence, and therefore the ratio of guilty to innocent within it differs from that of the general population. It applies the same analysis to the rule “the burden of proof rests on the claimant” and to the presumption that “what is in a person’s possession is his,” and argues that statistics about all objects in the world are not relevant to an object that is presently in legal dispute. It explains that the subgroup of objects under dispute does not necessarily preserve a ratio like “ninety-five to five,” and therefore the probabilistic argument would lead to the unreasonable conclusion that “usually the plaintiff is the liar.” It concludes that the justification for these rules is legal-institutional, such as prevention and deterrence so that it will not be worthwhile to seize someone and sue him without cost, and not decision-making on the basis of pure probability.
Venn Diagrams and the Representativeness Fallacy as a Formal Description
The text describes the fallacy through a Venn diagram: there is a large circle of the general population and an internal division by a general trait, and then there appears a small circle of a relevant subgroup that is positioned unevenly relative to that division, so that the ratios within it change. It defines this as a situation in which “what happens in the small circle does not represent what happens in the broader population,” and therefore applying the general majority to the subgroup is a representativeness fallacy. It notes that this is true both with the presumption of innocence and in monetary law, and that the gap between “what really happened” and “what is ruled legally / halakhically” stems from the fact that the rules are not built as a statistical majority but as framework considerations.
Rava, Rava’s Wife, Trust in Judges, and Maimonides’ Approach
The text tells of a workshop for judges in which the Talmudic passage was presented about Rava, who obligates a woman to take an oath, and then his wife tells him that she is a pathological liar, whereupon he reverses the oath against her. It describes the jurists’ objection to the idea that the judge’s wife could influence a verdict by whispering something to him, and in response argues that Rava knows the oath will lead to injustice, so naturally he should go with the truth. It formulates the tension as a choice between truth and formal rules designed to prevent a slippery slope and abuse, and argues that this depends on how much credit one gives one’s judges. It attributes to Maimonides, at the beginning of chapter 20 of the laws of the Sanhedrin, the principle that in monetary cases “the judge should do what his eyes see,” and that the essence of the ruling is conviction, not rigid rules of evidence, and notes that the Rif says that nowadays this was abolished because of the deterioration of the courts. It emphasizes that procedures are born of lack of trust, but their price is deviation from the truth, and that sometimes in order to prevent future problems we create present ones.
“The Majority Has Been Undermined” in Yevamot and Ketubot as Conflicting Majorities and a Subset
The text brings Yevamot 35a regarding a doubtful case of a child who may be a nine-month child of the first husband or a seven-month child of the second, and Rava’s question to Rav Nachman: “Let us say, follow the majority of women, and the majority of women give birth at nine months.” It quotes the Talmud’s answer, which adds another majority: “Every woman who gives birth at nine months, her pregnancy is recognizable after a third of her days,” and since here it was not recognizable, “the majority has been undermined.” After the Talmud’s difficulty, it quotes the correction: “Most women who give birth at nine months have a recognizable pregnancy after a third of their days,” so that this is one majority against another. In parallel it cites Ketubot 16a, where most women marry as virgins, but there is also the rule “every woman who marries as a virgin has publicity,” and one who has no publicity—“her majority has been undermined.” There too the Talmud corrects it to “most women who marry as virgins have publicity.” It explains that the structure is not “one majority canceling another” as some accidental mechanism, but rather a shift to a subgroup of women who have no publicity or of cases in which the pregnancy was not recognizable, where the proportions among the options are no longer the same as in the general population. It demonstrates with numbers that if the majority of virgins is 80% and the majority of those who “have publicity” is 80%, then within the group of “no publicity” one gets a majority in the direction of non-virgin, and if the second majority changes to 75%, then within “no publicity” it comes out even; therefore, without numerical data, the Sages treat this as a case that is not decided.
The PISA Example and the Claim that Almost Every News Number Falls into This Fallacy
The text uses the example of PISA tests and argues that public discussion takes a general number about the education system and applies it to the relevant economic question, whereas the relevant group is “one-tenth of one percent of the population” that is responsible for “ninety percent of GDP.” It argues that the distribution inside this small circle is not necessarily like the general distribution, and therefore the general number is not necessarily relevant to the correct consideration. It adds that there is almost never a time when one hears a statistical number in the news without falling into this fallacy or “its cousin,” because people take a general number and ignore information that places the case in a different subgroup.
A Rare Disease, a 99% Test, and a Fishing Net of the Wrong Size
The text presents the example of a rare disease with a prevalence of one in ten thousand and a test that is 99% reliable with symmetric error in both directions, and argues that someone who comes out “sick” may still have only a 1% chance of actually being sick. It calculates this by sending a million people for testing: one hundred truly sick people versus about 10,000 false positives among the healthy, so that the conditional probability within the group of “positives” is about 100 out of 10,000. It explains this with a fishing metaphor: in order to catch a rare phenomenon, the resolution of the test has to be on the same order as the rarity, so a 99% test is not “good enough” when the prevalence of the phenomenon is one in ten thousand. It emphasizes that the practical solution is that the test is not given to the entire population but to a subgroup with symptoms—that is, with “some supporting element”—which narrows the circle and raises the base prevalence.
The Roy Meadow Case, Crib Death, and Munchausen Syndrome by Proxy
The text brings the case of Sir Roy Meadow, where it was claimed that the chance of crib death is one in eight thousand, and therefore two such deaths are one in sixty-four million, from which murderous guilt was inferred and the mother was sent to prison until a statistician came and refuted the inference. It presents a possible claim of dependence between the events due to a genetic factor, so that probabilities cannot simply be multiplied, but says that this is not the main problem, because even one in eight thousand is a small probability that could still serve as a basis for conviction if the direction of the calculation is mistaken. It formulates the problem as an “inversion”: once the rare event has already happened and has reached the courtroom, one has to compare between two rare possibilities of roughly the same order of magnitude—“double crib death” versus “double murder”—and not conclude that one is correct merely because it sounds rare. It again compares this to the fishing net and to Bayes’ formula, and argues that within the “small circle” of families in which children have already died, the ratio between possible causes may be close to fifty-fifty, and therefore there is no basis for conviction from the general number alone.
99% Evidence in Criminal Law, Confession, and “Some Supporting Element”
The text tells of a conversation with a district court judge who said that 99% evidence is enough for conviction, and in response it argues that if the phenomenon being tested is very rare, like “murderers in the population,” such evidence is “worth nothing” when applied to a random person off the street. It applies this also to self-incrimination, and says that in general law there is a tendency to view confession as highly reliable evidence, but even that may still be insufficient if the prevalence of the crime is very low. It argues that in practice the system does narrow the group by means of prior indications such as location, circumstantial evidence, bloodstains, and so on, so that within the relevant suspect group the base rate changes and 99% evidence becomes meaningful. It explains that this also stands behind the requirement that “a confession is accepted if there is some supporting element behind it,” because that “supporting element” is a mechanism that narrows the space of possibilities and moves the discussion from the large circle to a smaller one.
Expected Value, the St. Petersburg Paradox, and the Probabilistic Critique of Pascal’s Wager
The text presents Pascal’s wager as an argument for keeping the commandments on the basis of an infinite expected reward versus a finite loss, and then argues that on the probabilistic level itself there is a mistake here in the decision criterion. It brings the St. Petersburg paradox: a coin-flip game in which the profit doubles in powers of two until tails appears, and the expected profit is infinite even though people would in practice pay only a small amount, and concludes that expected value is not the correct criterion when it is generated by rare tail events that have no practical chance of occurring in a one-time participation. It adds the example of a coin with a one-in-a-million chance of an enormous win, and concludes that it is not right to invest despite the high expectation, because expectation describes an average over many repetitions and not necessarily a one-time decision. It applies this to the atheist who says that the chance of God is one in a million, and argues that within that framework there is no decisive justification for the wager on the basis of expected value alone. It ends by saying that mistakes in the use of majority and probability can be either mistakes in calculating the probability or mistakes in interpretation and in the method of decision-making based on the calculation.
Full Transcript
[Rabbi Michael Abraham] Hello, today I want to touch on a topic that intersects with the previous topics I spoke about, which are basically probabilities in law and so on, but from a different angle. I hope today I finish this series—we’ll see. I’m never very good at estimating how much I’ll manage to cover.
[Speaker B] There are questions here—tough students.
[Rabbi Michael Abraham] I want to deal with biases. Daniel Kahneman calls this representativeness fallacies, or base-rate biases, or there are all kinds of names like that which are really very similar to one another. It doesn’t matter right now; we don’t need to get into fine distinctions. I’ll explain the basic principle in general terms. Maybe I’ll start with a current-events question. People often talk around Bibi, so very often they talk about the question of the presumption of innocence. After all, an indictment has now been filed against him, and the debate proceeds as usual between right and left, even though ostensibly that shouldn’t have anything to do with right and left. The debate is over whether the presumption of innocence still stands for him. When do you want him to resign? What do you mean—there is a presumption of innocence. The assumption is that as long as it hasn’t gone through the test of a court, you can’t convict a person. Now, in this matter there is a very, very simple distinction, and I feel it’s always sort of there under the surface and very often people aren’t aware of it. I’ll define it perhaps through an example, just to make the initial distinction. In the laws of slander, people often tell us that you may not believe it, but you should be concerned about it. Meaning, some rumor comes to me about someone I want to do business with, and they tell me, listen, this guy is a swindler, you can’t rely on him, such and such. The question is: what do I do with that? So in principle this is slander. As long as there are not two witnesses—after all, someone who knows testimony, a single witness who knows testimony, is not even allowed to go to a religious court, because on the basis of one witness you can’t decide the case, and therefore it comes out that he just spread slander, even though what he said is true. But he spread slander for no reason, because in the end it won’t be accepted, so he merely spread a person’s bad name in public. So in the context of slander too, they tell us: look, where it concerns you—like Jews, where it concerns your pocket—you may be concerned. Meaning, don’t believe it, be righteous, but if your money is at risk then you may be concerned. What does that mean? I think the simple meaning of this is that one has to distinguish between the halakhic plane and the factual plane. On the factual plane, if you ask me whether there’s a good chance this is true, certainly yes. Someone tells me this, and assuming he is not suspected of lying, then yes. On the other hand, there is some halakhic prohibition against believing it. What does it mean, a halakhic prohibition against believing? If it’s true, how can one forbid me to believe something that, in my estimation, is factually true? Fine, one can argue about that, but there is some halakhic principle saying that until you are certain to a very, very high degree, you may not accept it. But you don’t have to throw your money in the trash, and therefore you should be concerned. In effect the subtext here is the opposite. It’s obviously true—only you’re forbidden to believe it. “You may not believe it, but you should be concerned”—that sounds backwards. You should be concerned, but you may not believe.
[Speaker C] You’re benefiting from this information—what do you mean, you’re benefiting from information that came to you through a prohibition?
[Rabbi Michael Abraham] Okay, so what should I do? Because of that I now have to throw my money in the trash? “Fruit of the poisonous tree”—that’s what it’s called. On the legal plane in Israel, for example, there is no principled problem with fruit of the poisonous tree—they accept it as evidence.
[Speaker D] If the police brought this information,
[Rabbi Michael Abraham] What information?
[Speaker D] A person who did all kinds of things in the yeshiva or something—
[Rabbi Michael Abraham] No…
[Speaker D] Ah, of course.
[Rabbi Michael Abraham] Sure, what do you mean, why not?
[Speaker D] You’re not allowed to go to a religious court or something…
[Rabbi Michael Abraham] No, if he is one witness…
[Speaker D] He is one witness?
[Rabbi Michael Abraham] No, but a religious court is a judicial institution, not an investigative one. You go to the police so they’ll find another witness. You’re one witness, you want them to find the second witness—of course you have to prevent the acts. When you go to a religious court, you’re not going in order to prevent; you’re going in order to punish. To punish, you can’t punish with that kind of evidence. A religious court is not an investigative institution; a religious court does not conduct investigations. So this distinction between the factual plane and the halakhic plane is seemingly also what lies at the basis of the argument about the presumption of innocence, right? You may not believe what they say about Bibi, but you should be concerned about him. Meaning, the discussion whether to place him as prime minister does not necessarily depend on the question whether he is guilty and ought to sit in jail. If it’s not about money?
[Speaker B] What? If it’s not about money?
[Rabbi Michael Abraham] So here it’s not… not necessarily money, but there is still some public interest that someone like that not be at the head of the system—or whoever thinks so. It doesn’t matter, I’m not going to express a position here.
[Speaker B] But you introduced the idea of money.
[Rabbi Michael Abraham] Fine, נכון.
[Speaker E] There are other aspects. The issue of the Takana Forum isn’t about money either. Right, obviously. It’s harm to the public.
[Rabbi Michael Abraham] Yes, exactly. Money is just one kind of harm. But I want to look at this for a moment from a different angle, from a statistical angle. Let’s look for a second at the presumption of innocence. What is the presumption of innocence based on? Ostensibly people will tell you it’s based on the fact that generally speaking, all in all, human beings are usually innocent. There is a minority of criminals. So usually people are innocent. So until the court determines that this person belongs to the minority, why assume he belongs to the minority? And this is basically a consideration that is ostensibly statistical. Right? Let’s look at it from another context—for example, the burden of proof rests on the claimant. Right? There is a legal rule saying that if someone sues me, then the burden of proof lies on him. I have a basic advantage because I am holding the money. And if he sues, the burden of proof is on him. How do you explain such a thing? In the Talmud itself, by the way, there are all sorts of formulations: the presumption that what is under a person’s control is his, “we are witnesses,” all kinds of such formulations, the presumption that a person is not presumed to be a thief—the assumption is that what a person is holding is apparently his by law; he didn’t steal it. Is this statement a statistical statement? Ostensibly yes. If you do statistics on all the objects in the world, and see in which house they sit, most objects in the world sit in the house of their owners. Right, that is usually the situation. An object found in a house that does not belong to its owner is relatively rare—in other words, less common than an object sitting in its owner’s house. So ostensibly, the burden of proof resting on the claimant is really based on a probabilistic claim.
[Speaker B] Is there some consideration of time here? What? Until now he was okay? And that’s a presumption of fitness—until now he was okay, so presumably he continues… No, that’s an estimate and not statistics.
[Rabbi Michael Abraham] No, you’re talking about an original presumption. Fine, that’s a somewhat different discussion. An original presumption is something else. The question whether a person’s presumption of fitness is an original presumption or a clarifying presumption—that’s a different discussion. I don’t think this is an original presumption. But that’s another matter.
[Speaker E] There’s some public-policy enactment here—to deter people from trying to extract money from you unlawfully.
[Rabbi Michael Abraham] Fine, we’ll talk about that too. But right now I’m talking specifically about the statistical aspect. The question is whether what I said here—the statistics we make about objects found in different houses—can really serve as the basis for this principle that the burden of proof rests on the claimant.
[Speaker E] No, because in statistics you have to prove what the sample is. Whereas it seems to me that if a person was caught ninety-nine times as a thief and convicted and sat in prison, does he have the presumption of innocence the hundredth time? In my opinion yes. No.
[Rabbi Michael Abraham] Well, no, that’s something else. I’m saying… let’s say, so I’m telling you no. Fine, so in that case no. The question is whether in a case that is not like that I can rely on the statistics. You’re right, but I want to make a stronger claim.
[Speaker D] In monetary law we do not follow the majority. What? In monetary law we do not follow the majority.
[Rabbi Michael Abraham] Okay, we already talked about that last time, but that’s with respect to the law that—
[Speaker D] Here too we don’t follow statistics, after the majority.
[Rabbi Michael Abraham] No, but I’m talking about the question whether there even is a statistic here, before the question whether in monetary law we follow the majority or not. Suppose we do follow it, yes—according to Rav, who says that in monetary law we do follow the majority. By the way, there are some medieval authorities (Rishonim) who say that even according to practical Jewish law we follow the majority in monetary law, and what Shmuel said, that we do not follow the majority in monetary law, is only in a problematic majority, the majority that Tosafot—
[Speaker E] say that in Bava Kamma and elsewhere. So the question is basically… exactly.
[Rabbi Michael Abraham] Exactly. Meaning, the statistics we made on all objects in all houses are completely irrelevant. Among all objects in all houses, the distribution there is, say, ninety-five percent sit with their owners and five percent sit in other houses. That’s true. But the object I’m talking about belongs to a very defined and specific subgroup. It is an object that is under dispute. Among the objects that are under dispute—the subgroup of objects under dispute—is there also a majority there that the object belongs to the one holding it? That is very far from certain. In fact, understand what this means: it basically means that when two people argue, usually the plaintiff is the liar. Right? That is basically what it means. Usually the plaintiff is the liar and the defendant is telling the truth. Why assume that plaintiffs are usually liars and defendants are usually telling the truth? I see no reason to make such an assumption.
[Speaker B] I thought not long ago you said this is a preventive act—the burden of proof rests on the claimant.
[Rabbi Michael Abraham] No, that was a previous discussion, and I agree. But now I’m explaining why. Because the probabilistic explanation isn’t right. So then why the other explanations anyway? But why is the probabilistic or statistical explanation not correct? Because we’re applying a rule that is correct for the totality of objects, and we’re ignoring the fact that we have additional information. We have certain further information about the object before us, and that information narrows the group we are dealing with. Because the narrower group we are dealing with—objects that are in legal dispute—in that group it is not clear that the ratio found in the general group is preserved, that there too ninety-five percent of objects are in their owners’ houses. That is really not clear. On the contrary, I would assume it’s really not like that at all. Let’s say fifty-fifty, I don’t know. Assuming the chance that the plaintiff and the defendant are lying is equal, then I’d say fifty-fifty, perhaps. I don’t know whom you ought to bet on. But certainly not ninety-five to five. If you think of it in terms of a Venn diagram, let’s say I have some large circle of objects in the world. Okay. Now I say most objects are with their owners. So I put a line here. To the right of the line there is a small slice, and to the left of the line is the big part of the circle. Okay. So that line basically separates—I’m waving my hands a bit here—but the small part of the circle is objects that are not in their owners’ homes. And the big part is objects that are in their owners’ hands. Now I say: but my object belongs to a subgroup, not just to the general group, but to the subgroup of objects that are in legal confrontation. Now the question is whether this little circle of objects that are in legal confrontation—where is it located within the big circle? Because it could be that the big circle is like this, here is the line, and the little circle straddles the line equally, divided fifty-fifty. The line divides the little circle fifty-fifty. Meaning, the little circle is not in the center of the big circle, but off to the side. And that means that the ratio between the two parts within the little circle, within the subgroup that interests me, is not the same ratio as in the big circle. That’s why this is called the representativeness fallacy. What happens in the little circle does not represent what happens in the broader population. Okay. Now basically the same thing can be said about the presumption of innocence. In the presumption of innocence I’m basically saying this: most human beings, if you look at people in the street, most human beings are innocent, not criminals. And that’s true; there are such statistics. But those statistics are not relevant to the discussion when someone is accused. Why? Because he belongs to the group of people for whom evidence has already been found, who have already gone through some procedures—they are a subset. The court has not yet decided, but they have gone through quite a lot of the legal process already, and there still remains a basis; a very significant basis has been found—it was impossible to rule out the possibility that they are guilty. Meaning, there is justification for filing an indictment. Among this subgroup of people who have gone through these legal procedures, it is not at all clear—or rather quite clear that it is not so—that the ratio of innocent to guilty within this subgroup is the same as in the general population. Therefore, this is exactly the same phenomenon. So here too, beyond that, it’s not only the difference between the factual plane and the legal or halakhic plane, like “you should be concerned about him.” That is one difference. But where does that difference come from? Why indeed does the legal plane not necessarily take the factual plane into account? Because in the end you are talking about a group that is a defined subgroup. You have additional information about it. So why indeed do we maintain the presumption of innocence? Or why indeed do we assume that the presumption is that what is under a person’s hand is his, in the previous example I gave? For legal reasons, not for probabilistic reasons. In terms of probability it’s fifty-fifty, let’s say for the sake of discussion. But still, on the halakhic-legal level we assume it anyway, and in each context it requires its own explanation. The presumption that what is under his hand is his—of course, as people here said earlier—is a preventive mechanism so that you won’t just grab someone on the street and sue him for no reason without paying any price. So he says, yes, I can grab anyone on the street; in fifty percent of cases they’ll believe me, in fifty percent they won’t, so overall I’ll profit some of the time. In order to make sure it won’t pay for me to do that, they say no—the burden of proof rests on the claimant. Unless you bring evidence, we won’t believe you. But on the legal plane I won’t take that into account. On the legal plane I’ll assume the other person is right. Same thing with the presumption of innocence. On the legal plane there is a lot of logic in this—and it connects to the previous classes, where we talked about statistical evidence, say that fellow who at some point was in prison, if you remember, or the blue and red buses in the city. I said there is probabilistic evidence to convict someone, but for one legal reason or another we still do not convict him. There I argued that this itself was even a statistical consideration, but in the accepted sense these are legal considerations. And there is such a thing very often: when you ask me factually what happened, it’s one thing, and when you ask me halakhically what happened, it’s something else, or legally what happened, it’s something else, because there are legal considerations. Legal presumptions are not necessarily based on statistics, on a statistical majority. Yes, and this reminds me that I think we spoke about this once: I once led a workshop for jurists—judges, lawyers, and others. Rakover organized some kind of judges’ week, and he invited me once, and we studied there the Talmudic passage where Rava is sitting in judgment and a certain woman comes before him, and he obligates her to take an oath. Then Rava’s wife comes and says to him, listen, I know this woman—she comes into the court, and in those days the court was of course in his house, one of the rooms in the house. She comes into the court and says to him, listen, I know this woman, this woman is a pathological liar, there’s no point in making her swear, she’ll swear falsely. So what did he do? He reversed the oath onto the other side. He said that the other party should swear and collect. Okay? And the people there were really outraged. What do you mean? The judge’s wife walks into the court, says, oh, I know her, she’s a liar, and the judge decides the verdict in light of whispers his wife whispers in his ear? What kind of legal system is that? What kind of legal system is that? So I said to them, listen—again, in light of the distinction I spoke about earlier between facts and law. I said to them, suppose you are in such a situation, and suppose you know your wife, yes, and she speaks truth, meaning you have no suspicion that she is lying or anything. Now what do you want Rava to do? After all, Rava knows that if he lets this woman swear, she will swear falsely and take the money; she will remain with money that is not hers. But because of legal purism, you want him to say: yes, yes, but since this is my wife, I completely ignore the truth—I know this is the truth. And I know that what I rule here will be a false judgment. An injustice. I will in fact be doing an injustice. But because I’m not allowed to listen to my wife, then no? Why? The man knows that this is the truth; he knows that the truth is that this woman is a liar. There is no point in making her swear. So why should he do it? Now, of course, all this is a bit disingenuous, because obviously this opens the door to problematic things. Meaning, tomorrow morning your wife will come and say, nail that guy because he did such-and-such to me, or whatever. Meaning, it’s problematic to run a legal system that way. Therefore, I accept that there are sometimes legal considerations that tell you not to act according to the truth. But you cannot argue with the fact that this is the truth. Meaning, once again, this is the distinction between truth and legal consideration, only this time in the opposite direction. Meaning, Rava went with the truth, and the judges wanted him not to go with the truth, but for legal reasons to ignore the truth. And this, as I told you, depends on the question of how much trust you place in your judges. If you trust your judges, that is what I would want them to do. This is the truth, so why on earth should he leave the money with the person to whom it does not belong? Rather, what—you think they’ll do improper things. So here it already depends on what kind of credit you give your judges. In a society in which you can give your judges credit, I would want them to judge this way. By the way, this is Maimonides’ position. Maimonides took his rule from there, at the beginning of chapter 20 of the laws of the Sanhedrin. Maimonides says that this is basically the source, from the Rif, that in monetary cases a judge should do what his eyes see. You don’t need rules of evidence, you don’t need two witnesses, you don’t need anything. What you need is to be convinced. Once you are convinced that this is the truth, that is how you rule.
[Speaker E] And he also reversed the oath—he didn’t rely only on his wife.
[Rabbi Michael Abraham] Yes, but reversing the oath gives power to the other side, of course. Not for nothing the party defending himself always swears. Yes, right, it’s not completely straightforward.
[Speaker F] So maybe judges should be required to consult their wives before every verdict.
[Speaker E] Or female judges—today there are more female judges.
[Rabbi Michael Abraham] Female judges should consult their husbands? Consult with whoever they find appropriate. Maybe in that case it was his wife who knew the situation, but it doesn’t specifically have to be his wife.
[Speaker F] Why can’t a legal system be based on that kind of happenstance? Why not?
[Rabbi Michael Abraham] Why not? It easily can. The only question, of course, is how much credit you give the judges. If you fear they’ll abuse it or do problematic things with it, fine, then we enter into rules that fence us in, and we don’t really give the judge the degree of freedom he deserves. But if you think of a hypothetical situation in which I have some trust in this judge—why not? I want him to do the truth to the best of his ability, without binding him with rigid rules.
[Speaker F] And why don’t we have due diligence?
[Rabbi Michael Abraham] No, why? Why do you need procedures? You need procedures when there is no trust. That’s exactly the point. You don’t need procedures; you need to get to the truth. We are so used to thinking that procedures are necessary because we are used to suspecting our judges. But if I had trust in our judges, I wouldn’t want procedures; the procedures would only get in the way. The procedures would only get in the way. What are the procedures? All the lawyers make a living from procedures. Why do you need procedures? Go to the judge, tell him what you have to say, and he’ll tell you who is right—that’s all. Why do you need all the legal leverage and all this pilpul and formalism? Why all this? Again, I don’t have another patent, I don’t know how to do it better, but all of this stems from the fact that we don’t trust our judges.
[Speaker E] In that period, was there no situation where a judge disqualified himself because he believed another judge would do more justice?
[Rabbi Michael Abraham] Why not? It could be there was, I just don’t remember at the moment.
[Speaker E] It could be a kind of little trick by the first judge.
[Rabbi Michael Abraham] You mean he could pull a maneuver. Resign, bring another judge who would believe the wife of the first judge, who of course is disqualified from testimony because she is a woman. But in the end you have to reach some kind of result that does not proceed according to the ordinary rules of law.
[Speaker E] Fine, in any case, let’s get back to our subject. It’s a matter of trust, it’s a matter of the judge here knowing—he has to rule. Not true, and what if he does know?
[Rabbi Michael Abraham] And he does know—there, his wife told him, this woman—
[Speaker E] is a liar, she knows her.
[Rabbi Michael Abraham] Yes, but in all other cases he doesn’t know. I’m talking about the cases where he does. If he recognizes that he doesn’t know, I have no issue. I’m talking about the cases where he does know. What should he do when he does know? But that’s against the official rules.
[Speaker E] But with—
[Rabbi Michael Abraham] The rules also don’t guarantee that he knows; the rules aren’t free of errors either.
[Speaker E] In hearing, certainly not in direct sight.
[Rabbi Michael Abraham] Yes, because there we put it into the formal category that the judges themselves become witnesses.
[Speaker E] Yes, that’s something like what you’re suggesting. What? Yes, it’s something formal.
[Rabbi Michael Abraham] No, but the formalism here organizes something that really ought to be straightforward. If the judges themselves saw it, that cannot be worse than two witnesses coming and telling me about the reality. On the contrary. There was simply an initial thought not to do it because indeed it doesn’t fit the rules. So now we’ll bring it into the rules—they themselves are the witnesses.
[Speaker E] How is that different from one witness? There?
[Rabbi Michael Abraham] One witness comes—
[Speaker E] and persuades—that’s one witness, but he’s only one witness. Also one witness, also relatives, everything.
[Rabbi Michael Abraham] I claim that in fact in monetary cases—and on this many later authorities (Acharonim) claim otherwise, but I think that’s not correct—in monetary cases one can accept testimony of relatives. But you don’t accept the relatives’ testimony because they testified; rather, if you became convinced they are right. Because their testimony is disqualified—but if you became convinced they are right, how is that worse than when you become convinced on your own? What does that have to do with a witness? It is also forbidden for the judge to be related, and nevertheless I accept my own conviction, so why shouldn’t I accept testimony from relatives? Rav Shmuel Rozovsky in Kovetz Shiurim and everyone writes that no, but I don’t agree with that at all—in monetary cases. The Rif writes, the Rif writes, that nowadays, since we have been scattered among the nations and the courts are no longer what they once were—yes, a kind of nostalgia that I don’t know how accurate it really is—but today this is no longer done. Maimonides himself brings this, that today it no longer exists; it has already been abolished—you can no longer do this. There are certain procedures, and that is exactly a reflection of what we saw earlier: that today we are already in a situation where we don’t really trust so much, and we work by procedures and not… But understand, there is a price for this. The price is that we deviate from the truth. We go with procedures so there won’t be problems, but that itself creates very serious problems. Sometimes, on the contrary, in order to prevent future problems we now create definite present problems. Like the problem of the slippery slope. Yes, you do something wrong now so that in the future you won’t do something wrong. Wait for the future—do the right thing now.
[Speaker B] You already finished this with “a person may execute justice for himself.” That too goes against… like what you’re saying. A person is convinced, acts, and that’s a kind of judgment. And if in court there is no evidence, no one tells you that isn’t judgment. There too, same thing.
[Rabbi Michael Abraham] And again, it depends on the question of what kind of trust you place, of course. There it’s harder, because there you have to trust the person regarding his own money. Here I’m trusting the judge; he truly has no personal stake. Does he care whether this one wins or that one wins? So there it makes a lot of sense. We need to get used to the fact that many times our minds are already enslaved to legal rules and we have lost contact with plain common sense.
[Speaker E] From the standpoint of Jewish law, fruits—what’s it called?
[Speaker B] Fruit of the poisonous tree?
[Speaker E] No, the poisonous tree.
[Speaker B] A person bursts in while someone else is speaking…
[Rabbi Michael Abraham] No, no,
[Speaker E] From the standpoint of Jewish law, if you have evidence that was obtained through…
[Rabbi Michael Abraham] I don’t know of any such disqualification. If it’s evidence, it’s evidence. Afterward, judge him for what he did, if it was a transgression. I don’t think that in Jewish law there is any disqualification of that kind for evidence. Strictly speaking, here and there there can be some enactment and so on… Fine, so what is the claim actually? The claim is that we use a distribution that is correct for the broader group even though we have additional information that could narrow us down to a certain subgroup, and there the distribution could be different. This thing has two examples in the Talmud itself; these are two passages built in really the same way… one passage in Ketubot and one passage in Yevamot. The Talmud in Yevamot 35—the Mishnah says: an uncertain son. A woman was divorced, or her husband died, doesn’t matter, and she is pregnant. Now she gave birth after a certain amount of time—say, a month later she remarried, and then she gave birth a month later—no, sorry, seven months later, yes, seven months later. The question is whether this son who was born was born after nine months from the first husband or after a seven-month pregnancy from the second. Uncertain whether nine months from the first, uncertain whether seven months from the last; he must divorce her, the child is fit, and they are liable for a provisional guilt-offering. Meaning, we’re talking about someone who did levirate marriage with his brother’s widow—he rushed to do levirate marriage with his brother’s widow—and now suddenly the woman gave birth after seven months. So what happens now? If this son is a nine-month son of the first, then he is in fact… no, he’s not a mamzer; if he’s a nine-month son of the first then he is fit, but she violated the prohibition of his brother’s wife. Right? Meaning, the levirate marriage actually—there is no levirate marriage in such a case, because the deceased had offspring, right? And if he is a seven-month son of the second, then again he is fit, because if he is the son of the second, then the first really did require levirate marriage, and since that’s the case, he is the son of the second, so from the standpoint of the child there is no problem. The question is what happens with the woman. So it says: he must divorce her, and the child is fit. He must divorce her because of the doubt regarding the woman, but the child is fit—what practical difference does that make here? It doesn’t matter whether it’s from seven months or from nine. So the Talmud says: Rava said to Rav Nachman, let us say, “follow the majority of women,” and most women give birth at nine months. Why don’t we rely on the majority? Most women give birth at nine months, not at seven months, so therefore the assumption should be that this is a nine-month child of the first. So the Talmud says to him: this is what I meant.
[Speaker B] The Talmud said he must divorce her, even according to his view?
[Rabbi Michael Abraham] Yes, but that’s because of uncertainty, whereas here it would be definite. It says to him: this is what I meant, this is what I meant—most women give birth at nine months and a minority at seven, and every woman who gives birth at nine months, her fetus is recognizable by one-third of her pregnancy. And this woman, since her fetus was not recognizable by one-third of her pregnancy, the majority has been undermined. Meaning, there’s a rule saying that if a woman is pregnant, in the first third of the pregnancy—that is, within three months—the fetus is already noticeable in most cases. You can already see that she is pregnant in the first three months. All right? And therefore, since here we did not see that, after all we would not have allowed her to enter levirate marriage, so because of that the majority is weakened. So it is not true that the majority of women who give birth at nine months simply stands as is; it has been undermined. Why? Because there is a counter-majority saying that if a woman is pregnant then the fetus is noticeable, and here it wasn’t. So therefore, most likely—what?
[Speaker B] Three months—what does that have to do with nine versus seven?
[Rabbi Michael Abraham] That’s two, what do you mean? No, by a third of its term it becomes noticeable within that period. It doesn’t matter—noticeable within that period; leave the technical argument aside for now, the Talmud says it should have been noticeable.
[Speaker B] But then that too is statistical. Everything is statistical. No, the recognizability is also statistical.
[Rabbi Michael Abraham] Right, of course. That’s exactly what we’re talking about, there’s…
[Speaker B] And that one is stronger?
[Rabbi Michael Abraham] Not stronger; these are two majorities that cancel each other out. There is a majority in favor of the side that she gave birth for the first husband, because most women give birth at nine months, and there is a majority in favor of the side that she gave birth for the second husband, because if it had been for the first, it would already have been noticeable. All right? And therefore—what?
[Speaker E] So it remains…
[Rabbi Michael Abraham] Balanced, and therefore it is uncertain: he must divorce her, and the child is fit. It’s balanced. All right?
[Speaker E] That’s not true, that’s not true, it’s not balanced. Why not? Because if after one month she is in her third month, then from one month after the marriage we know, in most cases statistically, whether she is pregnant or not.
[Rabbi Michael Abraham] Yes, the Talmud asks exactly your question. If every woman who gives birth at nine months has a recognizable fetus by one-third of her pregnancy, then since this one was not recognizable by one-third of her pregnancy, it should be definite that he is a seven-month child of the latter husband, no? After all, you saw that it was not recognizable by one-third of her pregnancy, so that settles it—we’ve decided the issue. So why are these suddenly two majorities that cancel one another out? So the Talmud says: rather, say this—most women who give birth at nine months have a recognizable fetus by one-third of the pregnancy. Not all of them, but there is such a majority. And this one, since her fetus was not recognizable by one-third of the pregnancy, the majority was weakened. Meaning, this rule that by one-third of the pregnancy the fetus is recognizable is not absolute; it’s not true in one hundred percent of cases, it’s true in most cases. That’s one line of reasoning. The Talmud in Ketubot goes in exactly the same direction. The Talmud in Ketubot 16: the Talmud gets to a case where a woman is before us, right? There is some dispute with the husband about the ketubah—whether she is entitled to two hundred zuz or one hundred zuz, right? The question is whether she entered marriage as a virgin or did not enter marriage as a virgin. Now there is a majority: most women marry as virgins. The rule is that most women who marry do so as virgins. Right? In most cases. You reminded me of that majority. Ravina said: because it can be said that most women marry as virgins and a minority are widows, but on the other hand every woman who marries as a virgin has publicity, and this one, since she has no publicity, the majority has been weakened. Exactly the same structure as in Yevamot. On the one hand, I want to know whether she married as a virgin or not—those are the two options—and there is a majority. That’s on one side. On the other side, those who marry as virgins have publicity, and in her case there was no publicity. Exactly like the fetus being recognizable by one-third of the pregnancy, and in her case there was no publicity. Now since there was no publicity, there is a majority in the opposite direction, because most women about whom no publicity emerged apparently did not marry as virgins, since if she had married as a virgin there would have been publicity. Therefore the majority is weakened. And again the Talmud asks your question: if every woman who marries as a virgin has publicity, then let witnesses come and testify. Those witnesses are false—it doesn’t matter, the case involved witnesses there. The Talmud says instead: Ravina said, most women who marry as virgins have publicity, and this one, since she has no publicity, her majority has been weakened. It is not absolute that publicity goes out for all women who marry as virgins; it is only a majority, and therefore the two majorities cancel each other out. Now the later authorities start discussing: then why does this majority override that majority, and basically why do I care that it is only a majority? After all, we follow a majority too; why does it have to be absolute? There is some misunderstanding here. This is exactly the same structure as I described with the Venn diagram. Basically, think—for the second case it’s the same thing, the cases are completely parallel, so I’ll illustrate with the second case. Let’s say that the majority that most women marry as virgins is eighty percent, suppose. And among the virgins, publicity also emerges for eighty percent—same strength of majority. So let’s assume there were one hundred women who got married in the city over the last year. Of them, eighty were virgins and twenty were non-virgins. Now among the virgins, eighty percent had publicity—that’s sixty-four. How many had no publicity? Sixteen, right? Now a woman comes before me about whom there was no publicity, and I ask myself whether she is a virgin or not. So twenty women out of the hundred are those with no publicity who are non-virgins, and sixteen out of the hundred are those with no publicity who are virgins, right? So there is a majority in favor of the first direction, and therefore I should rule that the woman is a non-virgin, right? What happens if the second majority is seventy-five percent, not eighty percent? Then there are one hundred women who married, eighty of them are virgins and twenty are not virgins. Among the virgins, seventy-five—not eighty—have publicity. Eighty marry as virgins, but among the virgins, seventy-five have publicity. So I have one hundred women who married, eighty of them virgins, twenty not virgins. Of the eighty virgins, three-quarters had publicity—that is, sixty—and one-quarter had no publicity, twenty. Now a woman comes before me with no publicity; there are two possibilities: either she is a non-virgin, and that’s why there was no publicity—that’s twenty—or she is a virgin, and she belongs to this minority for whom no publicity emerged—that is also twenty. That is already an evenly balanced doubt, right? This means that it is exactly the same phenomenon as the presumption of the body, like a presumption that shifts from one matter to another. What happens? We apply the majority that most women marry as virgins to the general group. But within the women I have more information—there is information that no publicity emerged about this woman. Within that small circle, when I ask myself what the ratio is between virgins and non-virgins, it is not eighty-twenty; it could be twenty-twenty, meaning fifty-fifty percent. It depends on the strength of the first majority versus the strength of the second majority. And since the Sages, of course, didn’t run statistics, the Sages said: if there is a majority in this direction and a majority in that direction, then the layers cancel each other out, on the assumption that you cannot decide. That is basically the point. It is true that in principle, if we were doing statistics and we had the numbers and knew that here it was eighty percent and here it was eighty percent, I could decide that the woman is a non-virgin. There would be a majority of twenty against sixteen.
[Speaker B] And in a doubt, what do you decide—at fifty-one percent you decide?
[Rabbi Michael Abraham] A majority of fifty-one percent is also a majority.
[Speaker B] For purposes of resolving a doubt, for every matter…
[Rabbi Michael Abraham] In Jewish law, a majority is fifty plus. So what we see here, basically, is two more examples where again we apply the general distribution to a subgroup about which we have additional information, and somehow we ignore that information even though it could be very relevant. That information shifts the distribution; it is no longer eighty-twenty but fifty-fifty, because within the small circle I already know that I’m in the small circle, so why take the information that is true for the whole big circle? Here inside this small circle—it’s like the PISA math tests, where we’re located roughly somewhere between Zimbabwe and Indonesia, and then all sorts of accusations always come out against education in the State of Israel and so on. But we all know, after all, that the people who need mathematics are maybe a tenth of a percent of the population, and they are responsible for ninety percent of the GDP. They need to succeed in math, so check the achievements of the State of Israel in that tenth of a percent. There I’m not sure the distribution is the same. In the education system, ninety percent are total failures in mathematics. Fine, so they’ll be bagging groceries in the supermarket even without math—it’s not such a disaster. Right, maybe for general education, but it doesn’t interfere on the level of economic considerations. When you look at the small and relevant circle, the question is what the distribution is there. That is the important distribution. Anyway, this is always true, by the way—there is never a time when you hear some statistical number on the news and they don’t fall exactly into this fallacy or its cousin. Meaning, they take some very general number and apply it to the case before us when we have information saying that the general number is not necessarily relevant. Now I want to show this in a slightly trickier way, something similar but much more confusing. I assume there are doctors here, or lawyers, or something like that; presumably there’s someone here from one of those professions. So let’s describe a situation in which someone comes to a doctor and the doctor wants to check whether he has some disease, a rare disease. Okay? Let’s say, I don’t know, there are a thousand patients in the State of Israel with this disease, which is about one in ten thousand, okay? That is the rarity of the disease. So he sends the fellow for a test, and it turns out that there is a good test for this issue, a test that is ninety-nine percent reliable. Okay? So he goes and takes the test, and to his disappointment it turns out that he has the disease. What is the probability that he is sick? Again, what—
[Speaker B] What was the first number? What was the first statistic?
[Rabbi Michael Abraham] A thousand patients in the group, one in ten thousand are sick. The test is ninety-nine percent reliable.
[Speaker B] Ninety-nine percent.
[Rabbi Michael Abraham] Ninety-nine percent, right, of course. No, that’s not correct. That’s not correct—come on, do the simple calculation. He can go home happy and calm and not worry at all; there is virtually no chance he is sick. No chance. You can whistle peacefully and walk back home. Why? Look—suppose there are one million people in Israel, all right? Of them, one in ten thousand are sick, so let’s say there are one hundred actual sick people. Right? The prevalence of the disease is one in ten thousand.
[Speaker B] Is that before… did you formulate it for me? What? After he was diagnosed as sick there is a sample group…? Wait, wait, now I’m explaining.
[Rabbi Michael Abraham] In a few minutes, not now. Right now we send all one million for the test to make the calculation easier so that we won’t need Bayes’ formula. So look, it’s very simple. Send all one million for the test. All right? How many of them come out positive on the test?
[Speaker E] One hundred percent of the sick.
[Rabbi Michael Abraham] How many come out positive?
[Speaker B] Ten thousand times 0.99.
[Rabbi Michael Abraham] No, no, the error is symmetric in both directions. A sick person can test healthy, and a healthy person can test sick, otherwise it gets complicated. For the sake of the discussion, the error is symmetric in both directions. You sent one million people for the test. Of them almost no one is sick—there are one hundred who are, but that’s negligible, right? One percent of them will come out positive.
[Speaker F] Ten thousand, yes.
[Rabbi Michael Abraham] Ten thousand will come out positive. How many really are sick? One hundred. Meaning that if you tested positive, your chance of really being sick is one hundred out of ten thousand—one in a hundred, one percent.
[Speaker B] That’s if the test really works both ways. Yes, yes, I’m talking about both ways.
[Rabbi Michael Abraham] Okay, your chance of being sick is one percent. Now the test is ninety-nine percent reliable. How does this miracle happen?
[Speaker E] If the doctor thinks you have the symptoms, then…
[Rabbi Michael Abraham] That’s another discussion—you’re right to point that out. I took someone off the street and sent him for the test. So what lies behind this, actually? It’s the same phenomenon I described before, by the way, but here it is much more confusing. Statistics are something very confusing—I don’t know, our minds just aren’t built to think statistically. There’s already an institute over there, the Center for Rationality at the Hebrew University, that makes a living off that. What do we actually see here? What we see here is that when you go fishing, the size of the holes in the net has to be roughly the size of the fish you want to catch. Right? Or smaller. Right? If you make big holes, you won’t catch small fish; you’ll catch whales. Right? And here too—meaning, you have a phenomenon that is very delicate, rare, right? One in ten thousand in the population is sick. To catch it you need a net with roughly that same resolution. Meaning, if the size of the fish is one in ten thousand, then the size of the net also needs to be one in ten thousand. You need reliability of one in ten thousand, and then it will come out fifty percent. Even then you still aren’t sure he’s sick, but then it’s fifty percent—that’s already an order of magnitude that requires concern. That’s the order of magnitude. Okay? That is actually what happened here, and it’s something that confuses people a lot. Meaning—and this is the same phenomenon I was talking about before, because look: within the group of those who were tested and found positive, the distribution is not ninety-nine percent or one in ten thousand; it’s straightforward. Focusing on the subgroup and then imposing on it the distribution of the broader group—that leads to a fallacy. Let me give another current example of this. Do you know Munchausen syndrome by proxy? Do you know that amusing phenomenon? They once accused “the starving mother” here—you know, remember that affair? Some mother who starved her children and so on. In the end many argued that the diagnosis was that she had Munchausen syndrome by proxy. What does that mean? Munchausen syndrome is like Baron Munchausen—someone who wants to attract attention, to tell all kinds of made-up stories in order to attract attention. Munchausen syndrome by proxy is the attempt to attract attention through someone else, not through myself. For example, a woman wants to attract attention, so she makes her son miserable. Then everyone will pay attention to the family: look what a miserable family, they have a suffering child; and maybe she even takes devoted care of him, which only magnifies it, but it’s not always that—rather it’s the very fact that there will be some family attention. So there is such a syndrome. The one who formulated this syndrome was Sir Roy Meadow, a British doctor. And the story was as follows: a British woman came to court; two of her children died of crib death—SIDS, two of her small children died. The question was whether it was crib death. So Roy Meadow came to court and said: the probability of crib death is one in eight thousand. The probability that two children would die of crib death is one in sixty-four million. Very roughly speaking, in that order of magnitude. So obviously that is not what happened. Since the chance is one in sixty-four million, how did they die? She murdered them. This woman went to prison. You won’t believe it—this woman went to prison and sat there I don’t know how long, at least months if not more, until some statistician came and gave that judge a real dressing-down, and among others Roy Meadow too—they even considered revoking his title. To this day I’m not at all sure that there even is such a syndrome called Munchausen syndrome by proxy. It’s an invention of psychology or psychiatry. In any case, the statistician’s argument—I think this is what he argued in court—which is correct in itself, was that since we do not know the cause of crib death, it is entirely possible that the cause is some genetic factor, and if one child in the same household died for that reason then the second did too. But that doesn’t help. Even a probability of one in eight thousand is a very small probability. Are you going to put her in prison on a probability of one in eight thousand? Not one in sixty-four million. One in eight thousand is a terrific probability in criminal law—they’d buy that. So what have you gained? Right, you’re saying the events are dependent and not independent, so you can’t multiply the probabilities. Fine, then take a single event: the probability is one in eight thousand. The problem here is not that at all.
[Speaker B] The problem is the inversion, because take millions of cases, or in a million…
[Rabbi Michael Abraham] What is the size of the fish? You’re using a net that, let’s say, even if it’s one in sixty-four million—suppose there is no dependence between the events, let’s go all the way in Meadow’s favor, okay? There is no dependence between the events; you can multiply the probabilities. The chance of this happening is one in sixty-four million. Excellent. What is the probability that a mother would murder her two children? How many such women are there among all women? I have no idea, but it also seems to me to be something around that order of magnitude. This doesn’t usually happen, right? Meaning, the fish you want to catch are very, very small fish. The size of the holes in the net is supposed to be the size of the fish. Or in other words, if there is a one in sixty-four million chance that a woman’s two children would die of crib death, then among sixty-four million women in the United Kingdom it will happen once. And that one time has now come to court. How can you convict? If you pick someone off the street independently and ask what is the probability that two of her children would die of crib death, the answer is one in sixty-four million. But now you ask the reverse question: it already happened. Two children died of crib death. Now it comes to court. The court now asks itself: there are two scenarios—either two children died of crib death, and that is one in sixty-four million, or she murdered them, and that is around one in sixty-four million too. That is also a probability… yes. So once both possibilities have very, very small probabilities and they do not add up to one—as we already see, that is exactly the fallacy; if you are deciding between the two they should have added up to one—they do not add up to one. Both possibilities are of the same order of magnitude, and you cannot decide. The fact that one possibility is very unlikely does not make the other one more likely. So how does that help? That is the principal problem. The problem is not the statistical calculation or the dependence between the events—why did they multiply the probabilities? Don’t multiply the probabilities, don’t multiply the probabilities. Even if you do multiply the probabilities, even if you don’t multiply the probabilities, one in eight thousand—as I said—is a fantastic probability for conviction in a criminal trial. You still cannot convict, because it is not the correct probabilistic calculation. When you look at people whose children have already died of crib death, what is the probability within that situation—that is our small circle—what is the probability within this small circle that it was the result of murder or the result of crib death? It may be fifty-fifty or something of that order. Therefore you cannot convict. The fact that generally, in the whole population, it almost never happens that two children die of crib death—true, it doesn’t happen. So what? But once it has happened, you ask the question under the assumption that it happened: what is the probability that it was crib death? That is a probability question—Bayes’ formula again. So the point is basically the same as with the rare disease. Meaning, if you want to catch fish, you need a net whose hole size is roughly the size of the fish. And if you equip yourself with a net with terrific holes but you want to catch tadpoles, that won’t help, even if the net has a very, very fine resolution, because you still won’t catch tadpoles with it. You need to match the hole size to the size of the fish. That is the principle.
[Speaker E] What is the probability that that person is sick after the test?
[Rabbi Michael Abraham] I said one in ten thousand. One in ten—one in a hundred. One percent. In the data I gave there, one in a hundred. That there is a one percent chance he is sick.
[Speaker B] And the test isn’t superfluous.
[Rabbi Michael Abraham] It isn’t superfluous; before it was less—
[Speaker B] Before it was more.
[Rabbi Michael Abraham] In short, it doesn’t tell you very much. What happens? And here I come to your point. Look—so how do you deal with such a thing? I had a friend who was a district court judge, and we often got to talk about the laws of evidence, among other things. I told him about this case. I said to him: look, evidence with ninety-nine percent reliability comes before you—you’re a criminal court judge—well, what if evidence with ninety-nine percent reliability comes before you? So he says: of course, obviously, I convict the person. Ninety-nine percent is excellent. I said to him: look, but how many murderers are there in the population? How many murderers are there in the population? Are more than one percent of the population murderers? Far less, right? Meaning, if you have evidence that diagnoses a murderer with ninety-nine percent reliability, it is worth nothing. You cannot convict in a criminal trial. The same is true, by the way, for a confession, a person’s confession—self-incrimination. When a person incriminates himself—“a person does not render himself wicked”—so there too, in the legal world, we once discussed this; in the legal world it is accepted that the problem is really what seemingly emerges from Maimonides, that you cannot believe self-incrimination because perhaps he is one of the crazy ones or something like that. In Jewish law it is not like that, but that is the assumption. So there too, regarding confession, they say it has a very high level of reliability. A person who convicts himself—there is a ninety-nine percent chance he really is guilty. What is the probability that he is one of the crazy ones? Right—but what is the probability that he murdered? The probability that he murdered is also not more than one percent. So evidence like that, even if it is ninety-nine percent, is not necessarily enough to convict the person. You need to compare the rarity rate of the phenomenon, or the prevalence of the phenomenon, against the strength of the evidence, against the reliability of the evidence. Except that—and here this really connects to what I opened with—usually a person who comes to an investigation or comes to trial is not arriving tabula rasa. Meaning, they got to him somehow. He was in the area, bloodstains were found on his clothes, there are indications connecting him to the act. Now that is a completely different story.
[Speaker B] Why? Those are the ninety-nine-percent pieces of evidence.
[Rabbi Michael Abraham] No, those are pieces of evidence—no, those… I’ve taken—there are three people. Exactly. So you place him into a small circle within which the distribution is already entirely different. And there the ninety-nine percent works well. Meaning, suppose that the people who were even on the scene were only ten people in total who could have committed the murder. Okay? Because it’s Agatha Christie. So there were ten people in the house. One of them committed the murder, so it’s one in ten. Now if you find evidence at the level of ninety-nine percent, that’s already not bad when you have ten people. If you jump on some random person on the street without any indication that he is connected to the matter, then indeed evidence of ninety-nine percent is worth nothing. Nothing at all. Evidence of ninety-nine percent in a murder trial against somebody just walking around on the street is worth nothing. But if there are initial indications and now you apply evidence of that strength, that’s something completely different. And by the way, I think that in law, for example, self-incrimination is accepted if there is some additional matter supporting the confession. If a person confesses to committing a crime, they accept it if there is some additional thing that supports that confession. And that is exactly the point. Even though lawyers understand this, that is the calculation behind it. What stands behind it is that this additional supporting matter basically tells me: from now on, it is no longer the entire public of the State of Israel that is potentially accusable of this offense; you have narrowed it down to a thousand people. Okay? That is what the “additional matter” does. You narrowed it to a thousand people. And you are still in a problematic situation, but for our purposes it is critical, because if you have evidence at the ninety-nine percent level and you are talking about 1,000 people, fine, then it starts becoming more significant. Or 100 people, or 10 people—then it starts becoming much more significant. The same thing—one second—the same thing regarding a doctor: if someone comes to a doctor, the doctor doesn’t just send him for the test for no reason. There are some symptoms; and that is exactly the additional matter. Once there are symptoms, then within the group suffering from those symptoms, the probability is no longer one percent that you have the disease. Therefore it is fine and it usually works. But one must pay attention, because many times mistakes can still be made, as in the case of Munchausen syndrome. I just want to finish, in these few minutes—
[Speaker E] In the story—you just told it about the doctor—it seems to me that you said the doctor didn’t just send him for that test. Okay, I said he didn’t just send him for that test.
[Rabbi Michael Abraham] I said: if he sent him just like that—we’re now doing public health testing, sending the whole public for coronavirus tests, right, sending everyone… then indeed the test is worth nothing. But if there are symptoms, that is the additional matter. That is exactly the point. Why? Because you take the small circle and don’t look at the whole big circle, and in the small circle it is different. I’ll just finish with one more statistical context. This could have been much longer, but I’ll try to keep it short, because with this I simply want to conclude this series. You know Pascal’s wager? Why should one keep the commandments? In Jewish translation: why should one keep commandments? So he says: because the expected gain is infinite. Meaning, if there is a God and you kept the commandments, you have infinite reward; if you didn’t keep the commandments, you get hit, God forbid. If there is no God, then supposedly you just enjoyed life a bit more or suffered a bit more—not significant. So overall, reason says to keep the commandments. If there is a God, huge reward; if there is no God, you didn’t lose so much, not a big deal. The expected value is positive, and therefore it is worthwhile to keep the commandments. Now there are many refutations of this argument, but to my surprise I suddenly realized that Pascal, as one of the fathers of this field called probability, actually stumbled here in the probabilistic domain. Beyond the question of what such service is worth at all—there are all kinds of other claims—I’m now talking about the probabilistic aspect. So here, on this issue… do you know the St. Petersburg paradox? They usually teach it when studying investment—traders in the stock market or things like that, when one studies capital markets, they show you the St. Petersburg paradox. Someone offers you a lottery. I toss a coin. If it comes up heads, you get two shekels and it’s over. If it comes up tails, we toss again. If it comes up heads, you get four shekels and it’s over. If it comes up tails, we toss again. Heads—eight shekels; tails—we toss again, and so on. Each time it’s a power of two: 2, 4, 8, 16, and so on. How much would you pay for a ticket to this lottery? Just throw out an answer without thinking—not sophisticated, just what seems right to you immediately. How much would you pay?
[Speaker B] You don’t lose on this, do you?
[Rabbi Michael Abraham] No, because you pay—you can lose the ticket price. The question is how much you would pay for it. Five shekels.
[Speaker E] Two shekels. The sum of a series?
[Rabbi Michael Abraham] No, after three times he wins—if there’s no other competitor then maybe that could be… huh?
[Speaker E] The sum of a geometric series, one-half plus one-quarter plus one-eighth, and that’s a series whose sum is…
[Speaker B] The sum of an arithmetic series is infinity.
[Rabbi Michael Abraham] No, no, no—the sum of that geometric series converges, but this is not the sum of the series. In a moment. You’re summing the probabilities; the probabilities of course add up to one, always, all possibilities together. I want to sum the payoffs, not the probabilities. Look, people would pay, I don’t know, at most 10, 20 shekels, roughly speaking, right? What is the expected value in this case?
[Speaker B] Infinite.
[Rabbi Michael Abraham] Infinity. The expected value is infinite. There is a probability of one-half of earning two shekels, probability of one-quarter of earning four shekels, probability of one-eighth of earning eight shekels. Right? So each such option is actually one shekel in expectation: one-half times 2 plus one-quarter times 4 plus one-eighth times 8—one plus one plus one, infinitely. The expected value of this lottery is infinite. So were we wrong in saying we’d pay only 10 or 20 shekels for it? And the answer is no. No, absolutely not. Why? Because the criterion of expected value is the wrong criterion for a lottery like this. We are used to thinking that the price of something is determined by its expected gain or expected loss. Not true. Expected value is a good criterion only where the expectation is something likely to occur, where there is a high probability that it will occur. But where the expectation is generated by very, very, very rare events, there is no chance in the world that you will receive the expected value, even though that is indeed the expectation. Let me give you an example. I toss a coin, but the coin is unfair. Meaning there is a one-in-a-thousand… no, there is a one-in-a-million chance it lands heads. Everything else is tails. Fine? But if it lands heads, you win ten to the hundredth power dollars. Fine? How much money do you invest in this lottery? Zero. Zero. Right? Because it won’t happen. But the expected value is enormous. Ten to the ninety-seventh or ninety-fourth power.
[Speaker D] Like the lottery, where you get a hundred million dollars with a very small probability.
[Rabbi Michael Abraham] No, and there too the expected value is low. The idea here—here it isn’t; the expected value is high. The expected value is high and still it is not right to invest. Why? Because expected value is not the correct criterion. In a place where the expectation is caused by tail events, by cases that have no chance of occurring—
[Speaker F] Exactly.
[Rabbi Michael Abraham] Meaning, if you repeat the event—if you repeat this lottery infinitely many times, then it’s worth it. Exactly. Meaning, to invest each time you have to buy a ticket. But if time after time after time you can participate in this lottery forever and you have infinite money, then what is correct to invest is infinity. But if you do it only once, even though ostensibly it is the same thing—what difference does it make? Here too you invest per participation, right? But no. If you do it only once, don’t invest a penny. It is worth nothing. Just like if someone offers me to outfit a ship and sail to some island in the Pacific Ocean—maybe there is a treasure there worth billions and billions of dollars. So how much does it cost to outfit a ship? Fine, a million dollars. What could happen? But the probability is tiny. True, the probability is tiny; multiply the probability and increase the expected value. That’s a start-up. Of course. Exactly. These start-ups—those ninety-nine percent that don’t work. The point is that very often the criterion of expected value is the wrong criterion. Expected value is a good criterion either where you do it many times—and much depends on how spread out the distribution is, how many times is “many”—or where it really lies near the center of the distribution more or less, and there is a reasonable chance that this will indeed be the event that happens. And it is also the event most likely to happen; not just the average profit, because that isn’t the same thing. Okay? Therefore the criterion of expected value is not correct. And in that context, if I return to Pascal’s wager, when some atheist comes—and this is how Pascal talks with the atheist. The atheist says: the probability that God exists is one in a million; there’s no chance. True, but if there is a God, you will gain infinity. So the criterion of expected value says to go with it. But that is not correct. Because according to the atheist’s own view, this is the kind of distribution for which it is not correct to go by expected value. And I just inserted this here to complete the picture; this is another statistical bias. In this case, by the way, it is not a computational bias. The previous things I discussed were simply mistakes in the calculation of probability. Here the calculation is correct. That really is the expectation. But making decisions based on the expectation is not correct. Meaning, the mistake is in your decision-making, not in the probabilistic calculation. The probabilistic calculation is correct, but the decision-making—meaning, in other words, our mistake in using a majority can be either a computational mistake, or a mistake in the question of interpretation—what this calculation tells me—or how to make decisions on the basis of that calculation.