Doubt and Probability—in Halakha, in Jewish Thought, and in General—Lesson 44 – Rabbi Michael Avraham
This transcript was produced automatically using artificial intelligence. There may be inaccuracies in the transcribed content and in speaker identification.
🔗 Link to the original lecture
🔗 Link to the transcript on Sofer.AI
Table of Contents
- The puzzle of the first child
- Statistics and assessing reality
- The connection to Monty Hall and the doors puzzle
- Rabbi Chaim of Volozhin’s claim about two majorities
- Dividing a negative majority into sub-majorities
- Can you get married without a religious court
- A negative majority versus a majority not directly before us
- The importance of wording the question in statistics
- The connection between the conditions of the question and the answer in cases of a judge
- Rabbi Goren’s example and two majorities in Kfar Etzion
- Understanding a majority directly before us in the Kfar Etzion situation
- The concept of an overwhelming majority and endless waters
- Three key questions: what do I know, what don’t I know, and what do I want to know
Summary
General Overview
The speaker opens with two anecdotes that lead into a broad discussion about probability, critical thinking, and the difference between persuasive numbers and relevant numbers. He criticizes a Home Front Command graph claiming that lying on the ground greatly reduces the risk of being hit by shrapnel, and stresses that there is no point in presenting foolish data just because it looks scientific. From there he moves on to explain conditional probability, the unintuitive nature of Bayesian thinking, and the need to ask questions precisely, especially in the laws concerning permitting agunot and in the use of “two majorities.”
The Home Front Command and the misleading graph
The speaker describes a Home Front Command notice stating that the closer you are to the ground, the lower your chance of being hit by shrapnel, and illustrates it with a graph showing 100% injury if standing, 85% if crouching, and 10% if lying down. He argues that the graph is absurd, because it creates the impression that tens or hundreds of thousands of people have already been killed by missiles, and even suggests that the problem is not merely bad wording but concentrated stupidity. In his view, it would have been better simply to say that lying down is safer than standing, without presenting misleading numbers.
Holy lies and critical thinking
The speaker says that people sometimes use “holy lies” to influence public behavior, even at the cost of creating a stupid world. He refuses to justify such a lie, even if it might increase discipline or fear of Heaven, and prefers correct statements that do not rely on stupidity. He sees the Home Front Command example as an important lesson in the need to use critical thinking and not automatically surrender to numbers and statistics.
Conditional probability and the two-children puzzle
The speaker moves to a puzzle about a woman with two children, one of whom is a boy born on a Tuesday. He first explains the simpler version: if you know only that at least one of them is a boy, then there are three equally weighted possibilities, and therefore the probability that the other is a girl is two-thirds. He then adds the information that the boy was born on a Tuesday, and recalculates the possibilities to arrive at the result 14 out of 27, that is about 51.8%. He emphasizes that the information about the day of birth changes the probability, even though the change seems almost negligible and is not intuitive.
Bayesian thinking and the intuitive difficulty
The speaker explains that the puzzle demonstrates how hard Bayesian thinking is for human beings—that is, updating probabilities in light of new information. He mentions Kahneman and Tversky and the distinction between System One, fast intuition, and System Two, conscious and orderly thinking. He says that in many statistical problems intuition misleads us, and therefore you need to write things down, reconsider the data, and not rely on gut feeling.
Additional examples of conditional probability
The speaker gives further examples in which people confuse two opposite questions: the quality of a judge versus the reliability of a court ruling, and the reliability of a medical test versus the probability that a person is actually sick. He explains that a test or a judge can be very accurate, and yet the result given to a specific individual can still be weak or misleading if the base rate in reality is low. The conclusion is that the right question and precise wording matter more than the numbers themselves.
Permitting agunot, two majorities, and a negative majority
The speaker moves to the topic of permitting agunot and explains the distinction between a single majority and two majorities. He says that Rabbi Chaim of Volozhin and the halakhic decisors who followed him permit certain cases on the basis of two majorities, especially when dealing with a negative majority—that is, where there are no precise numerical data, only a general assessment. He emphasizes that if the majority is a positive majority with clear numbers, then what matters is the percentage itself, not whether it was formed by multiplying majorities.
The mistakes of Rabbi Yitzchak Elchanan and Rabbi Goren
The speaker criticizes the use of two majorities by Rabbi Yitzchak Elchanan and Rabbi Goren. Regarding a ship that sank, he says it is meaningless that most of the passengers were non-Jews, because the husband of the woman in question was certainly Jewish, so only one majority remains: most people who fall in drown. Regarding Kfar Etzion, he argues that there too one cannot speak of two majorities, because the married men were a minority, and captivity or death do not create a sufficient multiplication of overwhelming majorities. He stresses that the question must be asked about the specific individual, not about the population as a whole.
Precision of the question and halakhic responsibility
The speaker keeps returning to the point that the most important thing in statistics and in Jewish law is to ask the right question. He argues that most errors stem from imprecise wording: what do I know, what don’t I know, and what exactly do I want to know. In permitting agunot, he says, the judges do not change reality but decide according to the information, and the woman herself can sometimes assess the situation, yet in practice it is accepted to go to a religious court in order to receive an orderly and responsible ruling.
Full Transcript
So
[Rabbi Michael Abraham] We’re in the middle of probabilistic multiplications, but I can’t help myself—I just have to open with two anecdotes, one of which really made me happy and amused me, and the other is just a nice anecdote. Right now Uriel has just joined, so I was literally about to start speaking in your praise while you weren’t here, and now here you are. Uriel sent me today—this was today, right?—he sent me some Home Front Command notice, a Home Front Command notice that simply gave me enormous pleasure. Here, you see?
[Speaker D] The more
[Rabbi Michael Abraham] The closer you are to the ground, the lower your chance of being hit by shrapnel. Therefore you should lie on the ground and protect your head with your hands, according to the explicit instructions of the Home Front Command. Remember, Home Front Command instructions save lives. So far, fine, although my son already wrote that unfortunately there are a lot of people who say, “It won’t happen to me,” and the truth is that most of them are right, because it really won’t happen to the overwhelming majority of them. Never mind. But the beginning is okay. But look at this amusing graph. They’re comparing here between three situations: if a person is standing, if he is crouching, and if he is lying down. So if he is standing, he has a 100 percent chance of being hit. If he is crouching, he has an 85 percent chance of being hit. And if he is lying down, he has a 10 percent chance of being hit. Therefore, in an open area you must lie on the ground. QED. It’s mathematics, you can’t argue with that. So fortunately there are people who stay alert, because the truth is that the press is full of these kinds of nonsense—you can find several examples every day—but this is really a wonderful example, because this is a formal notice from the Home Front Command. Meaning, it’s not some journalist who got confused about something. That made it especially amusing. So in short, what this means is that whenever a missile flies toward the State of Israel, everyone who is standing dies. Our population ought to be pretty depleted by now, in light of the last few days, because everyone who stood died, everyone who crouched—85 percent of them died—and everyone who lay down, 10 percent of them died. Wow, we’ve lost tens or hundreds of thousands of people in the last few missiles, and for some reason the news forgot to tell us about it. You understand this is simply nonsense in tomato sauce. I mean, even if you try very hard to defend some hidden intention behind this notice, I think it’s hard to make it work. It’s not even just bad wording—it’s simply concentrated stupidity.
[Speaker F] But they meant Bayes’ theorem: given that the person is standing there where the explosion is, then—
[Rabbi Michael Abraham] Given that he is standing where the explosion is, he is 100 percent dead? That’s also not true. Unless the missile actually hit him. Okay, but if the missile hits him, he’s 100 percent dead whether he’s lying down, standing, or crouching. In short, it’s nonsense. I don’t know—I tried to think of some model where a shard is flying parallel to the ground, say at human height, between zero and one meter seventy, one meter eighty—then if the person is standing, it hits him 100 percent of the time, right? If the person is crouching at, say, 85 percent of his height, then 85 percent chance it hits him; and if he’s lying down, then 10 percent of his height, so 10 percent chance it hits him. But that’s only if there is exactly a shard flying parallel to the ground at a height between zero and one meter seventy and exactly along the path where the person is—in other words, it crosses the perpendicular axis where the person stands.
[Speaker G] Rabbi, rabbi, rabbi—
[Rabbi Michael Abraham] Basically—
[Speaker G] The rabbi is suggesting that if they had done it the way the rabbi says, then they’d say it like this: if you stand, your chance is 0.000-something percent of being hit. If you sit, another zero. And if you lie down, add one more zero. Put that on the graph, and no one in the State of Israel goes to any safe room, because he says, “If my odds are that tiny, I’m not going into any safe room.”
[Rabbi Michael Abraham] So what I’m saying is: the advised solution is either to lie—holy lies, so to speak—which is what they did here, under the optimistic assumption that this is a holy lie. In my opinion, the idiot was the one who wrote the notice, not that he lied to people because he thought they were idiots. But maybe, I don’t know—that’s my guess. The better advice is not to present a stupid graph—not this way and not that way. Don’t present graphs. Say that it’s safer to lie down than to stand. That’s all. And that’s a perfectly correct statement. Everything is fine. If they had written only that introductory sentence, everything would have been fine. The introduction is fine. But we have some kind of—we’re captive to—this is a very important lesson, I think. It makes me happy that you noticed this because of our course, because that really is one of the things I hope for. I mean, to activate our systematic critical thinking a bit about all sorts of things floating around us. There is so much nonsense floating around us, and I assume 95 percent of the population who saw this graph didn’t blink. Everything was fine. Our tendency is to use numbers, because numbers are facts—you can’t argue with numbers, it’s certain, it’s persuasive, and so on—and therefore they keep feeding us statistics and numbers. Now, 98 percent of them—look, I just made a statistic too—98 percent of the statistics you see in the media are nonsense. From the non-quantitative survey I’ve done, from my non-quantitative impression, I’ve almost never found—maybe once, very isolated cases—that someone brought a statistical datum that was actually relevant to the claim he wanted to make. It almost never happens. Sometimes there’s a certain relevance, but even then you can challenge it in many ways. The level of irrelevance can vary. Sometimes it’s not relevant at all, sometimes it’s partially irrelevant, sometimes it’s relevant but not entirely convincing, and certainly not numerical or quantitative or precise. It just doesn’t happen. It really doesn’t happen. I mean, when someone brings a number that is actually the right number for the point he wants to make—that just doesn’t happen, there’s no such thing. There may be numbers that have some connection to what you want to claim—that’s the best case—and that does happen sometimes, and sometimes there is no connection at all. It’s just nonsense. So we are so used to worshiping numbers and supposedly scientific facts that they take us captive. Now there is something very healthy in wanting to rely on solid, quantified facts, yes? Science is built on that, and very good—that’s a healthy and good approach. Again, not to be captive to it, but certainly to try to stick to solid and quantitative facts. That’s definitely a welcome stance. The problem is that it leads us into nonsense. Because what happens is that the moment someone presents numbers, he wins. Even though the numbers can be nonsense. If you insist that the numbers be correct, then sticking to numbers is very good; it helps decision-making and thinking and everything. But the moment you see a number, from your point of view it’s good, from your point of view it’s true, and that simply leads to the opposite. It leads to stupid decision-making. Therefore— Rabbi? Yes.
[Speaker G] If, say, the rabbi were, Heaven forbid, commander of the Home Front Command—because then we’d lose him to that, so that’s why I say Heaven forbid—if the rabbi had to issue instructions to the public, would he write: “Look, the truth is there’s no logic in going to the safe room because the risk is very small, but there’s Kant’s categorical imperative, so please obey the Home Front Command instructions”?
[Rabbi Michael Abraham] You’re going back to what you said before, and I’ll go back to the answer I gave you before. I wouldn’t write that. I wouldn’t write that, but I also wouldn’t write a lie.
[Speaker G] So what would the rabbi say? Would he say the truth? That is the truth, after all. The truth is very unpersuasive, very hard to motivate the average person with.
[Rabbi Michael Abraham] Why? I’d tell him: go into the shelter because the shelter is safer, and people will get hurt if they don’t go into the shelter. That’s all. And that’s a true statement.
[Speaker G] But the probability—no—
[Rabbi Michael Abraham] I’m not getting into the question of what the statistical chance is that a specific person will be hurt. What does that have to do with anything? I definitely would not make false and stupid statements in order to influence people. I would not do that.
[Speaker G] Yes, on that I agree with the rabbi. I understand—you also have to understand the—
[Rabbi Michael Abraham] That’s the dilemma of holy lies. Holy lies. In the religious world it’s very common to use that technique. I mean, they lie to you so that you won’t entertain thoughts of heresy, Heaven forbid. They tell you, I don’t know, that all the Amoraim could revive the dead and had divine inspiration, and they could never make mistakes. The fact is that the Talmud has quite a few mistakes in it, but if you repeat it enough times no one notices. Everything is fine. They were all fiery heavenly beings, they never erred, they revived the dead morning and evening, they were ministering angels. Okay? Now why—what is the justification for telling this crude and foolish lie? The justification is that then all of us will stick to what the Talmud says and be righteous and do what is right. Now, I accept that that may indeed be the result. That probably will improve commitment to the laws of the Talmud if we educate people on this nonsense. But I’m not willing to pay that price. I’m not willing to educate people through stupidity in order to produce fear of Heaven. Falsehood does not stand—not only in the sense of evaluating reality, but also because falsehood itself is not something that stands. Not because it won’t last and people will discover it’s a lie. No—even if they never discover it. To stand on a lie is not called standing. In other words, I don’t want faith that is based on stupidity, even if I can manage to create it. Now here it’s a bit different, of course. You’re not trying to create faith, you’re trying to save people’s lives. So what do you care if you lie to them a little in order to save their lives? And here I’m really not unequivocal. I mean, I’m willing to listen to people whose policy would be, yes, let’s lie to people if it improves their chances of survival. I’m willing to hear that. But it seems to me that in this context it’s just ridiculous. Don’t present the graph; tell people, listen, inside the shelter it’s safest, shelters save lives—completely true statements. Everything is fine. Without getting into how many lives it saves and what the probability is for a specific individual and things like that. You don’t need to present that. It doesn’t matter. These are true statements. You don’t need to resort to lies, even if perhaps the lies would improve things slightly. But I think at this level it’s not worth the lie, because there is also a price to creating a stupid world. Creating a stupid world also has costs. Therefore I think it’s wrong to pay them.
All right, that’s the first example. The second example is a puzzle. I happened to see some video on YouTube—Presh Talwalkar, for those who know, Mind Your Decisions or something, I don’t remember—videos about mathematics. And it’s always very nice, not at very advanced levels; I think most people can understand most of the videos there, and it’s wonderful. He really does beautiful things there. Simple, clear, very nice. I mean, there’s nothing like taking the dog for a walk and watching a Presh Talwalkar video. In any case, he gives a puzzle there. A woman comes to you and says she has two children. One of them is a boy—not only one, rather there is one who is a boy, there could also be two—and he was born on Tuesday. What is the chance that the other one is a girl? Not related to the previous example; this is another statistical anecdote. What is the chance that the other one is a girl? At first glance it looks like—what do I care about the first one? The second one is something, and there is a 50 percent chance he’s a boy. Let’s say the distribution of boys and girls is 50-50, then the chance is 50 percent. Right? The first is a boy, you know that; the second you don’t know—either a boy or a girl, equal odds, 50-50. Not true. The chance is not 50-50.
[Speaker E] You have more information, so it has to change the result.
[Rabbi Michael Abraham] No, not every additional piece of information changes the result, only relevant information. The question is whether this information is relevant. What do you say? It’s not a half—I already told you, I already gave you a hint. Not a half.
[Speaker G] What? How is it not a half?
[Rabbi Michael Abraham] Well, that’s exactly what I’m asking you. Ah—the fifty—
[Speaker G] Percent for boys is spread over seven days, but now it’s only Tuesday.
[Rabbi Michael Abraham] Let’s go step by step. Forget Tuesday. A woman comes and says she has two children, one of them is a boy. Another five days—
[Speaker H] So you count all the possibilities. You have first child boy, second child boy, that’s one possibility; boy-girl, and that’s it. Boy-boy, boy-girl. Okay, so what’s the chance?
[Speaker I] It seems unconditional, no? What do you mean unconditional? I mean, the identity of the sex of the second one doesn’t seem connected to the question of what the first one’s sex was.
[Rabbi Michael Abraham] So you’re saying it’s 50 percent.
[Speaker I] Yes, I’m asking, at least. Does everyone agree?
[Speaker G] No, no, but wait—again, we’re talking about children who have already been born.
[Rabbi Michael Abraham] Yes, yes, she has two children.
[Speaker G] But if it were about what will be born now, then maybe apparently it’s 50-50. But if you say two children and the chance is 50-50 and one has already been born, then indeed the other—the chance that it will be a boy originally was 50-50.
[Rabbi Michael Abraham] No, it’s independent. What does that have to do with anything? Birth is independent, just as you said before, there’s no connection.
[Speaker G] No, but they were both already born.
[Rabbi Michael Abraham] I’m asking—if one child was born and he’s a boy, and they ask me what’s the chance that the other will be a girl? Fifty percent. There is no dependence between the two children. So what I said here is something else. She says she has two children, at least one of whom is a boy, and the question is: what is the chance the other is a girl?
[Speaker I] At least one, okay, that’s like with that choice there, I don’t remember the name of it. The sheep, two sheep and the car—Monty Hall.
[Rabbi Michael Abraham] No, that’s—yes, Monty Hall. Look, if she has two children, there are four equally weighted possibilities, right? Two boys, two girls, boy and girl, girl and boy. Agreed? Each of those has equal weight, 25 percent. Okay? Now she says one of them is a boy. That means the possibility of two girls drops out. So what’s possible here is either boy-boy, boy-girl, and girl-boy. Right? Three equally weighted possibilities. You didn’t say the first one is a boy, okay.
[Speaker H] Right? You didn’t say the first one is a boy?
[Rabbi Michael Abraham] No, no—one of them. I didn’t say the first. Right? So those are three equally weighted possibilities. Right? In one of the three possibilities the second is a boy, and in two of the three possibilities the second is a girl. So the chance is two-thirds.
[Speaker G] Wow, that’s really disturbing.
[Rabbi Michael Abraham] Right. Wait, we haven’t gotten to Tuesday yet. Tuesday is even crazier. So the chance is two-thirds. That’s how it is, it’s simple, right? And that is exactly what Michael, I think—Michael said before, that when information is added, it can change the probabilities because this is conditional probability. I’m asking what is the chance that the second is a girl given that I know one of them is a boy. Okay? So that’s not the same as the chance that this child is a girl, which is the unconditional probability. Now, of course, as I told him, the question is whether the first datum—the thing on which I condition the probability—is relevant information. In the calculation I just did I showed you that yes, it is relevant information. Therefore the addition of the information changes the a priori probabilities. This is the posterior versus the prior, yes, the prior. The prior is one-half, but the posterior is two-thirds. We talked about conditional probability.
[Speaker I] Wait, so if they had said the older child is a boy, that would change the picture, right? What? If they had said not one of them is a boy, but the older one is a boy, would that also give the same picture of the probability?
[Rabbi Michael Abraham] She has two children and the older one is a boy? Yes? No, then it’s 50-50.
[Speaker I] Ah, then it’s 50-50, yes, okay.
[Rabbi Michael Abraham] Now Tuesday. Now she says not just that I have two children and one of them is a boy, but I have two children, one of whom is a boy and he was born on Tuesday. What is the chance the other is a girl? We have added more information, right? That he was born on Tuesday. The big question is of course whether this information is relevant, because otherwise irrelevant information does not change the conditional probability.
[Speaker E] Was it that same child who was born on Tuesday?
[Rabbi Michael Abraham] Yes, yes, that boy was born on Tuesday.
[Speaker E] Okay.
[Speaker H] Then in the same way, wouldn’t you make all the possible combinations of boy and girl born on each of the seven days, and then—
[Rabbi Michael Abraham] So what do you say—how do we do it? Let’s see. We have—we already saw that there are three possibilities, right? Boy-boy is one possibility, girl-boy and boy-girl, right? Let’s look at the possibilities for boy-boy. How many possibilities are there for boy-boy in terms of days? Let’s say the chance of being born on any of the days is equal, yes? That’s my assumption. So one boy was born on Tuesday. Then there is the possibility that it’s the first boy and the possibility that it’s the second boy, right? If the first boy was born on Tuesday, then there are seven possibilities for when the second boy was born, right? Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Sabbath. Okay, that’s seven possibilities. If the second boy was born on Tuesday, how many possibilities are there for the first boy?
[Speaker J] Also seven, no?
[Rabbi Michael Abraham] Apparently also seven, but here is the trick. It’s six. Because if you say that the first one was born on Tuesday, we already counted that possibility. Twice, yes. Right? So basically we have 13 possibilities for the birth of the two boys, right? If they were born on the same day, then both on Tuesday. If they were born on different days, then you have another 12 possibilities. So altogether it’s 13 possibilities. Okay? That’s on the boy-boy side. In other words, if the second one is a boy, there are 13 possibilities. What happens if the second is a girl? Then we have the possibility of girl-boy, where the boy was born on Tuesday, which we know, and there are seven possibilities for when the girl was born. And the possibility of boy-girl, that’s also seven possibilities: the first boy was born on Tuesday, and seven possibilities for when the girl was born. Here there isn’t the overlap that there was there, right? So how many possibilities are there that the second is a girl? Fourteen. Fourteen. So altogether, out of 27 possibilities, there are 14 possibilities that the second is a girl. Which means that the chance that the second is a girl is 14 divided by 27. 51.8 percent. Now notice, this is fascinating. First of all, it’s not 50 percent but 51.8. If I had just told you that answer, you’d pull your hair out. How do you get 51.8? Everything here looks so nice and round—half, I don’t know, two-thirds I can still somehow swallow, but what is 51.8? Yes, it’s 14 divided by 27. But more than that, notice that the added information that the boy was born on Tuesday reduced the chance that the other is a girl from two-thirds, 67 percent, to 51.8, almost to 50 percent. In other words, it almost completely canceled all the change that happened from the information that one of them is a boy. The information that he was born on Tuesday almost canceled all the effect we got from the information that one of them is a boy.
[Speaker I] Wait, and even before—
[Rabbi Michael Abraham] Before that information we almost got back to 50 percent, almost.
[Speaker I] But even before that information I could have said he was born on some day—wouldn’t that have brought us to the same result?
[Rabbi Michael Abraham] Everyone was born on some day.
[Speaker I] Yes, I mean, what—does Tuesday help us more than Monday or Sunday or any other day?
[Rabbi Michael Abraham] No, Monday is the same too. But “some day” is not. If you know the day, it doesn’t matter which day. It could be Tuesday, it could be Sabbath, it doesn’t matter.
[Speaker I] I mean, if I assume the day but I don’t know it, then not?
[Rabbi Michael Abraham] Very, very, very unintuitive, right? It doesn’t make sense—why should the fact that he was born on Tuesday change this story at all? Fine, regarding the two-thirds we were already convinced, but this Tuesday business is bizarre. But it’s bizarre in both directions. Once I’m already convinced that the first answer is two-thirds, I’d say, so what if he was born on Tuesday? It’s still two-thirds.
[Speaker I] But what—
[Rabbi Michael Abraham] What bothers me is that whatever day he was born, we’d get the same result. But on the other hand it’s also not a half. Meaning, it’s not two-thirds and it’s not a half—it’s surprising in both directions. It’s surprising that it’s not two-thirds, and it’s surprising that it also didn’t go back to half. It’s something that is almost half. In other words, the information that this boy was born on Tuesday almost cancels out what the information “one of them is a boy” did to us, but not completely. It still leaves us with another 1.8 percent change. These are the wonders of conditional probability, and it really is something very unintuitive. But this is exactly the kind of confusing play of conditional probability.
[Speaker G] What is this? The rabbi enjoys messing with our minds with this, it’s just—
[Rabbi Michael Abraham] Why? It’s so much fun. What could be nicer than this? A total loss of common sense.
[Speaker G] What? A total loss of common sense.
[Rabbi Michael Abraham] On the contrary, it’s straightening out crooked reason. Common sense is often crooked, and in order to make it straight you need to straighten it. Not every time I think something is it common sense. When I think something, that’s my first glance. But the first glance is not always common sense. This is what Daniel Kahneman spent his whole career on, and Tversky as well—the point that our first glance is very far from common sense. We have very serious problems with statistical thinking. There are various evolutionary explanations for why we get so tangled up with statistical thinking and conditional probabilities. All Bayesian thinking—the idea that if I have information, I need to reorganize the data in order to understand the probability of each possibility—that’s called Bayesian thinking. And we have a hard time with Bayesian thinking. Our intuition doesn’t get along well with Bayesian thinking. We have to do it in an explicit way, write it down with formulas, not let our intuition run wild. Here Kahneman’s System Two has to work, not System One. Kahneman said that System One is the unconscious intuition, the automatic pilot, which often leads me to more efficient and quicker behavior than conscious behavior. Therefore many times, say, in battle or something like that, instinctive reactions can keep you alive, whereas if you stop to think for a moment, there will soon be nothing left for you to think about. But in problems where our intuition is not good, where it is not properly built, there conscious thinking leads us to better results. Systematic thinking—go with pen and paper, not with intuitions. And in statistical Bayesian thinking especially, this is very common. People make mistakes every step of the way, and sometimes the mistakes are critical. We’ve seen quite a few examples of this during the course. One second. For example, if you remember Munchausen syndrome, with that woman whose two babies died and they said she murdered them because crib death twice was impossible. And I said there that the mistake was that they did not compute the conditional probability properly. Or the difference between—if you remember when we talked about majority in a religious court, I said that when I say, for example, that I have judges and I measure their quality, let’s say the parameter p determines the quality of the judge. The quality of a judge, p, say 0.9, between zero and one. Let’s say it’s 0.9; that means that in 0.9 of the cases he is right. In other words, if X happened, the chance that the judge will determine that X indeed happened is 0.9. So the judge is of quality 0.9. Now I have three judges, each of whom is of quality 0.9, and there’s a two-to-one ruling in some case. Two say that Reuven murdered someone and one says Reuven did not murder. So what is the chance that this ruling is correct? We did some calculations there—you remember I also referred you to an article by Nadav Shnerb—and I said that what confuses people is that the question I just asked is the reverse of the first question. The measure that assesses the quality of the judge asks the following question: given that X happened, what is the chance that the judge will indeed determine that X happened? In other words, given that Reuven murdered, what is the chance that the judge will determine that Reuven really murdered? That determines the quality of the judge. A good judge, of course, whenever Reuven murdered, will determine that Reuven indeed murdered, right? That’s a good judge. So the measure of the judge’s quality is the following conditional probability: given that Reuven murdered, what is the chance that the judge will rule that Reuven murdered? But the question I’m asking about the ruling is the reverse question: given that the judge ruled that Reuven murdered, what is the chance that he really did murder? If the first is P of A given B, this is P of B given A. And since there are very few murderers, and we saw the easy arithmetic involved here, there can be a dramatic difference between these two probabilities. The judge can be a judge of absolutely marvelous quality, 0.99, and the chance that he correctly identified the murderer can be, I don’t know, one percent. In other words, he is almost certainly wrong. Rabbi?
[Speaker G] Yes. I’m still troubled by the answer. Suppose a woman gave birth to one boy and she knows it’s a boy, and now she is pregnant and asks the doctor—let’s say before the ultrasound era; it’s a scene in Kishon’s film Abu el Banat, if the rabbi remembers. He asks the doctor, “Tell me, what are the chances I’ll have a boy?” and it’s a bit more than 50 percent, and he insists on saying no, I’m sorry, it stays 50 percent. Was Kishon wrong there or not?
[Rabbi Michael Abraham] No, not at all. Fifty percent. He’s right.
[Speaker G] So what’s the difference, basically? The difference is that here you know the first one is a boy.
[Rabbi Michael Abraham] The information that the first one is a boy is irrelevant information, so the conditional probability equals the prior. What’s the difference now between a half and two-thirds? The difference is whether the information is irrelevant—in that case the probability stays one-half. When A and B are independent, then P of A given B is the same as P of A.
[Speaker G] But what’s the difference? I asked the rabbi before: if the child has not yet been born and the woman asks us what the probability is, then the rabbi says 50 percent. But if he has already been born, the rabbi says it’s no longer 50 percent?
[Rabbi Michael Abraham] Why? Why? Not if he has already been born—no. Even if both have already been born and you tell me the first was a boy, the chance the second is a girl is a half. It has nothing to do with whether they have already been born or will be born. The question is what information you are given. If they tell you the first is a boy, then the chance is a half. If they tell you one of them is a boy, then the chance is two-thirds.
[Speaker G] Why? What’s the difference? Why should I care? The order doesn’t interest me. So why is it two-thirds?
[Rabbi Michael Abraham] Because if they tell you the first is a boy—if they tell you the first is a boy—you have more—
[Speaker E] More information from the outset when they tell you it’s the first one. If they tell you it’s at least one of them, regardless of order, then you have more options. Think about the calculation I did.
[Rabbi Michael Abraham] I said there were four groups.
[Speaker G] Ah, I understood the calculation, but from the—
[Rabbi Michael Abraham] On the other hand it contradicts intuition. Right. So if they tell you the first is a boy, there are only two groups, not three: boy-girl and boy-boy. That’s it. And if they tell you one of them is a boy, there are three groups.
[Speaker G] So biologically it didn’t change the situation? No.
[Rabbi Michael Abraham] Because the question isn’t biological, the question is statistical. Nothing here has to do with biology; it has to do with my information about reality. Obviously what I determine regarding what is likely in reality is a function of what I know. If you know something else, then you will determine something else about the same reality, and both of us will be right. Statistics does not determine what reality will be; statistics says what my best estimate is, given the information I have about reality. So if you and I have different information, our answers will be different and neither of us is mistaken. That’s right—they really should be different, because the estimate is a function of the information. Why is insider trading forbidden? Because if you have inside information, you’ll win 100 percent of the time. And someone who doesn’t have the information won’t win. Why? Reality is the same reality—either the stock went up or it didn’t. Yes, but he has information and you don’t.
[Speaker H] But when you count all the possibilities like that, it’s basically because you’re in a symmetric space. In other words, the possibility of having a girl and the possibility of having a boy is identical. If it weren’t 50-50, we couldn’t do the calculation this way?
[Rabbi Michael Abraham] We would do the calculation accordingly, obviously. I did say that I’m assuming equal probability for boy and girl.
[Speaker H] And do you have an explanation for why it supposedly contradicts intuition? Even though—right, it contradicts intuition—but is there some explanation for it? There are evolutionary explanations. I mean, why our intuition—
[Rabbi Michael Abraham] Fails in Bayesian thinking.
[Speaker H] No, I mean more mathematical explanations—more to understand where our mistake is, so to speak. Why is the perspective not correct, for example?
[Rabbi Michael Abraham] Our mistake is that we think this information is irrelevant. We think the conditional probability equals the absolute probability. The posterior equals the prior. And that’s not true. This information is relevant.
[Speaker H] So basically someone who has really internalized the point of conditional probability supposedly should not be surprised by this?
[Rabbi Michael Abraham] Someone who has really internalized it—or someone who is careful to use pen and paper when thinking about questions like this. If someone has really internalized it, then maybe his intuition will already work correctly. But even if you haven’t internalized it, if you know this without internalizing it and you use pen and paper and don’t immediately blurt out the intuitive answer, you’ll probably also reach the correct result.
[Speaker H] And would you also say this is the same as that puzzle with the two doors, where once they tell you, listen—
[Rabbi Michael Abraham] Monty Hall? I did indeed talk about that in one of the classes at the beginning of the series.
[Speaker H] It’s basically the same thing.
[Rabbi Michael Abraham] What do you mean the same thing? There too our intuition falls—
[Speaker H] Because of conditional probability.
[Rabbi Michael Abraham] Right, there too it’s a problem of Bayesian thinking, of conditional probability. Most of our problems in statistics are Bayesian thinking. Somehow evolution doesn’t support Bayesian thinking. I don’t understand why, but people have written about it. Meaning, there’s quite a bit of material on why we fail at Bayesian thinking. Okay.
[Speaker G] If he had given us information not about his birthday, but about his height, his eye color, his weight—would that affect it?
[Rabbi Michael Abraham] It could. It depends—you have to see what assumptions you’re making. Say it’s his birth date, so that’s relevant because you know there’s an equal probability of being born on any given day. If they give you information about his height, and there’s an equal probability that he’ll be within some range of heights, then it’s the same thing. Yes.
[Speaker K] There’s no doubt that in Gehenna they study statistics.
[Rabbi Michael Abraham] And in heaven they study statistics. It’s just that for some people, apparently, heaven is Gehenna. I always—I never tended to disbelieve those who say, “Look, math just isn’t for me, I can’t understand it.” I always thought that was just prejudice. The person is hopeless in advance, it somehow looks hard to him, so he doesn’t manage with it; but if he weren’t despairing ahead of time, he actually would manage with it. I don’t know, that’s my feeling. But you know, evolution doesn’t support Bayesian thinking, so maybe I also need to draw conclusions in light of the data I’ve been exposed to over the years and arrive at the conclusion that after all there are people for whom this is harder. But I don’t know. Well, it’s certainly part of it—part of that is definitely true. Meaning, for many people it’s a kind of fixation: from the outset they somehow decide this isn’t for them, it’s complicated, it’s Gehenna like Shmuel said, and so they recoil and don’t really try to understand it, they give up in advance. That’s definitely part of the phenomenon, in my view. Meaning, improving a person’s confidence will certainly improve performance. And still, apparently people do have different inclinations, and some people are more mathematically inclined and some less so, apparently. Okay.
[Speaker G] But I know there’s an approach—a psychologistic approach, this approach, I don’t know, in philosophy—that logic too is psychologistic; basically, we’re built that way. I know the Rabbi doesn’t agree with that, and I also tend against it, but there is such an approach, and serious people have said it.
[Rabbi Michael Abraham] Well, that just shows that serious people can also be wrong. Okay, so let’s get back to our topic—we had a little fun. It’s like Rabbi Akiva, right, who would make up riddles in order to wake up the students. “Why did Esther merit to reign over one hundred and twenty-seven provinces? Because she was the descendant of Sarah, who lived one hundred and twenty-seven years.” Right, that sort of astonishing midrash of Rabbi Akiva. Okay, so we’re in the topic of probabilistic products. I want now to talk about—we’re dealing with permitting agunot through two majorities. I’ll perhaps mention in one sentence: the claim of Rabbi Chaim of Volozhin, and many halakhic decisors who follow him, is that although a single majority is not enough to permit an agunah, two majorities, or a majority of a majority, yes, I didn’t distinguish between a majority of a majority and two majorities—a majority of a majority does help. And I said that maybe the source for this is the law of “water with no visible boundary” itself, which is already from Talmudic law. In water with no visible boundary, basically, most of those who fall in drown. Right? Most people who fall into the water drown. But such a majority is not enough to permit the woman, at least rabbinically, and therefore Rabbi Chaim of Volozhin says that if there is another majority, then yes. For example, what I suggested is that “water with a visible boundary” is really Rabbi Chaim of Volozhin’s example. Water with a visible boundary basically means that the person who fell in probably drowned. And even on the chance that he didn’t drown, if he came out then I would have seen him, because I can see the shore in water with a visible boundary. So there are two majorities here. Once there are two majorities, I permit. But if there is one majority, I do not permit. I said that the Chazon Ish does not accept the permit of two majorities at all, and we explained why. Right, I said that basically the Chazon Ish says: what are two majorities? It’s just a stronger majority. So what? Why does that matter? If one majority were ninety-six percent, what difference is there between two majorities yielding ninety-six percent and one majority yielding ninety-six percent? The question is what the percentages are. And the answer I gave to that, in the name of those who do accept two majorities—and this is a very important point that will continue accompanying us later, which is why I’m repeating it—is that the majority we’re talking about here is a negative majority and not a positive majority. A negative majority means I do not have the numerical data—apropos numbers, right? We do not have numerical data about the strength of each majority. Take water with no visible boundary: most of those who fall in drown. What does that mean—how many? Ninety percent? Ninety-eight percent? Seventy percent? How many? I don’t know how many. I have an assessment that most drown. But I don’t have numerical data about how many drown. Okay? Only in a case like that is there room to distinguish between a majority and two majorities, between a majority and a majority-of-a-majority. Because if the numerical data are known, then majority times majority gives you a number—namely ninety-six percent. Rabbi.
[Speaker E] And now—
[Rabbi Michael Abraham] Then you have to decide whether ninety-six percent is enough to permit or not. The difference between a majority-of-a-majority and a majority—the difference between a majority-of-a-majority and a single majority is stated only in situations where the majority is a negative majority. But then, I explained that I want to be sure that the husband really is dead, and my only problem is just a formal halakhic problem of the laws of evidence, that I don’t have witnesses or something like that. But I need to be sure that in reality the husband is dead. To be sure the husband is dead, I need an overwhelming majority or an absolute majority. A simple fifty-one percent majority is not enough, because no halakhic decisor would ever think of permitting a woman to remarry if there’s a fifty-one percent chance her husband is dead. There’s a forty-nine percent chance he’ll come back. No one would dream of permitting a woman to remarry in such a situation. It makes no sense to follow the majority in a case like that. So why do we permit in the case of water with a visible boundary? Because the husband will not come back—we know that—it’s an absolute majority. So basically the claim is this: when I have a negative majority, I can still divide it into a number of levels: an ordinary majority, an overwhelming majority, and an absolute majority. An overwhelming majority is like water with no visible boundary: it is almost certain that he drowned, it is very rare that he was saved, but there is still some chance. Here, on the Torah level, the woman is basically permitted. An overwhelming majority is enough on the Torah level. The rabbis say no, an overwhelming majority is not enough; we want an absolute majority. How do we establish an absolute majority if we are dealing with a negative majority? If it were a positive majority, you would say ninety-nine point eight percent, okay, you could set a numerical threshold. But when the majority is a negative majority, we have no way to verify that this is an absolute majority. So what do we say? We want it to be a product of two overwhelming majorities. Meaning, if most of those who fall in drown, and someone who comes out—there is a majority that I would see him in water with a visible boundary—then there is here a product of two negative majorities, which already comes out enough, it comes out as an absolute majority. On that basis I can permit. But only because the majorities are negative does it matter that I multiplied majorities here. If the majorities are positive majorities, then I will still permit the woman, but there I will permit her on the basis of a number. I don’t care whether it is a product or not a product. Tell me what majority you want to reach, ninety-nine point eight? Fine. If there is one majority of ninety-nine point eight, the woman is permitted. If there are two majorities whose product gives ninety-nine point four, then the woman will not be permitted, because it isn’t ninety-nine point eight. Therefore it is not relevant there to ask whether this is the product of two majorities or one single majority. When we talk about a positive majority, what speaks here are the numbers. And it makes no difference whether you reach that number by multiplication or directly. The multiplication is relevant only where the majority is a negative majority, and then you have no way to set the threshold. What are you going to say, ninety-nine percent? I don’t have percentages; I don’t know what the percentages are. So I say: okay, if it seems overwhelming to you, and you can multiply it by another majority that is overwhelming, then you can treat it as an absolute majority, and for you that is like certainty. Like a person who came and saw the identifiable face with the nose—there too there is some chance he was mistaken, but as far as I’m concerned that’s certainty. I mentioned that this is connected to what we said about statistical evidence: that there is a difference between knowledge I have and a statistical estimate I have, even though apparently they are the same numbers. Yes.
[Speaker G] Can I ask a somewhat more preliminary question? What force does the statement of the halakhic decisor, the judge, or the Talmud even have, that she is permitted or forbidden? This is a matter of reality. If the husband is alive and he shows up, then it doesn’t matter what they say. Obviously. She is permitted to remarry, and we do not know the reality. No, but they tell the woman: the chance that you still have a husband is very, very small. She has to decide for herself. What is this statement at all? The Sages say: we decide.
[Rabbi Michael Abraham] Everyone can decide for themselves, but the judges are the ones who are supposed to make such decisions.
[Speaker G] The judges decide for her whether she is allowed to remarry.
[Rabbi Michael Abraham] But they can’t—
[Speaker G] Change the facts.
[Rabbi Michael Abraham] They don’t change the facts. But if she says, “I’m sure—”
[Speaker G] “And I’m going to remarry…”
[Rabbi Michael Abraham] Suppose I told you that a scientific researcher or statistician would determine this—would you accept that? Right? Because he has more skill in how to assess reality.
[Speaker G] He can give me the numbers—what does he know?
[Rabbi Michael Abraham] No, he has to tell me the statistics. He has more skill in how to assess reality. He can’t give you numbers, but he knows how to assess reality better than a layman.
[Speaker G] But if she is an authority, a researcher, she won a prize… she isn’t an authority?
[Rabbi Michael Abraham] The ordinary citizen is not an authority. The ordinary citizen would make the same mistakes that Rabbi Herzog made, and he had a doctorate. You can’t rely on the ordinary citizen to interpret the—
[Speaker C] Future.
[Speaker G] But in the end, is it just some side advice? That’s what I mean to say. It’s side advice—if she’s a researcher and an expert in statistics, she’ll say: I checked the data, I’m a researcher, I conduct surveys—
[Rabbi Michael Abraham] Then let her go to a religious court and get married without asking them. Fine, it could be.
[Speaker A] But she can’t reduce—
[Speaker C] No, exactly, you need the formality of it so that she is no longer a married woman.
[Rabbi Michael Abraham] What?
[Speaker G] No, not formality—she won’t be a married woman.
[Rabbi Michael Abraham] She is forbidden to live with someone. If the woman—suppose the husband, suppose the husband died in water with a visible boundary, okay? Now the woman goes and remarries; she doesn’t ask the religious court. She studied Talmud too. She’s not a judge, meaning, it’s not a ruling of a religious court, but she knows the situation, and she knows that in water with a visible boundary she may marry. She can’t marry without—of course she can; if a man comes and gives her a perutah, she’ll be married.
[Speaker A] No, absolutely not.
[Rabbi Michael Abraham] Friends, let me say this for a moment, let me say it. So she goes and gets married without asking the religious court. Fine? She took the risk, made her own calculation, everything is fine. Now what happens in a case where it isn’t water with a visible boundary, it’s something else—but still the woman knows how to assess probabilities, and she says: I’m going to marry because as far as I’m concerned this is like water with a visible boundary. After all, the judges would make the same calculation. The judges too—it’s not always a case of a person falling into the sea. There are other cases where we say: is this similar to water with a visible boundary, is this similar to water with no visible boundary? So the judges too are actually making some assessment of reality, and so the woman says: I too will make an assessment of—
[Speaker L] Reality.
[Rabbi Michael Abraham] Fine, fine.
[Speaker A] Again, we’re speaking here within the framework of Judaism—you can’t, no one can make any assessment and get married without first being permitted. Why not?
[Rabbi Michael Abraham] Why not? What do you mean why not? Of course she can.
[Speaker A] What do you mean, of course she can? If she accepted betrothal from a man without going to the religious court and informing them, she cannot accept betrothal from someone else.
[Rabbi Michael Abraham] What do you mean—of course she can, he’s dead. Huh? If he’s dead, he’s dead in the house, he’s dead. He’s completely dead. Can she marry afterward without going to a religious court?
[Speaker A] How does she change her status from married to widow?
[Rabbi Michael Abraham] He died, he died in her house. Ah, in her house, okay. But still she needs to go to a religious court to get permission? No—
[Speaker A] For that, no. Why not? But in the above case that we’re discussing… why not? We’re speaking here about the case… what do you mean why not?
[Rabbi Michael Abraham] Why not? She changes status—how can you change status not in a religious court?
[Speaker A] You can’t.
[Rabbi Michael Abraham] Well, but here she changed status.
[Speaker A] Again, in the case where there is testimony of death in her home—
[Rabbi Michael Abraham] What testimony? She was at home and her husband died before her eyes; nobody else saw.
[Speaker A] Then she does have to go to the religious court.
[Rabbi Michael Abraham] What are you talking about? What are you talking about? She can get married, good health to her. What do you mean? She knows she’s unmarried and she goes to get married. More than that: the credibility of one witness in permitting a woman is based in part on the presumption that a woman checks carefully before marrying. A woman who goes to get married is exacting and checks carefully, because she knows how badly she’ll get into trouble if she marries and then her first husband comes back.
[Speaker C] Wait, Rabbi, even—
[Rabbi Michael Abraham] The religious court’s ruling itself is actually based on the woman’s own checking.
[Speaker A] Right, but here in this case, if no one knows he died, she won’t be able to remarry, because they’ll ask her.
[Rabbi Michael Abraham] One second. No one knows he died—she knows the same data the religious court knows. That’s what she knows. She knows he fell from the ship into the water and no one saw him come out. She knows that; the religious court doesn’t know more than she does. That’s what we know. But she has to go through a religious court in order to marry. Why?
[Speaker A] Does she have to go to a religious court to notify them that he died? No.
[Rabbi Michael Abraham] Then how will they receive testimony from her in order to marry her if she—
[Speaker G] They don’t need testimony in order to marry her off; she gets married on her own.
[Rabbi Michael Abraham] What—you mean for registration, for civil registration, at the Interior Ministry, the rabbinate, I don’t know what—there they’ll ask for testimony, because from their point of view they are taking action. But the woman herself—if you ask whether she can go get married, let her go get married, good health to her. If she comes to ask, then the religious court will have to sit and deliberate and make decisions. Or if the religious court has to register her in the community ledger or in the residents’ registry book or whatever it may be, then they’ll do it according to their procedures. But the question whether the woman can go get married—she can do whatever she wants.
[Speaker C] Rabbi, if so then Shmuel’s question is entirely in place. What? If a woman knows, she’s capable, and she manages to decide as a religious court decides—she has all the data and all the knowledge—then she too can decide to marry without a religious court.
[Rabbi Michael Abraham] Correct, and the calculation she has to make is the same calculation the religious court would make: with two majorities and water with a visible boundary, water with no visible boundary, everything is fine—let her make that calculation. And let her understand that there’s a difference between a tree like this and Rabbi Herzog’s tree and all the implications we drew here. One of the reasons people nevertheless do go, and customarily go, to a religious court is that they want some decision from a body that is skilled in this kind of thinking. So one doesn’t want every person making decisions for himself, because most people may make mistaken decisions. But this is not a matter of principle. It’s just something that became established because it’s safer. But there’s no principle here that says you absolutely have to do it. I don’t see any reason. Is there no issue of formality, of official change of status? No. That’s what—
[Speaker A] I said. Just as if he died in her house and she saw it—
[Rabbi Michael Abraham] That too is a change of status. She can go get married; she doesn’t need approval from a religious court. But again, I’m speaking about official status. Like, something… I’m saying there is no official status in Jewish law. That’s what I’m saying.
[Speaker E] From the standpoint of Jewish law, let her marry, that’s all. What is official status? In any case, in the state when they do registration there is status, and then there are procedures—
[Rabbi Michael Abraham] And then it goes through the religious court. That’s all fine. That’s because we want to do it in an orderly public way. If you’re asking whether a woman is allowed to go get married, let her get married. In any event, genealogical records were written that way, that’s how all the Jewish people behaved. A genealogical registry is like the population registry here. Fine. Obviously when there is a public body managing it, it manages it according to procedures. But the woman herself—if she did it properly, then she did it properly. Okay, in any event, back to our subject—I… that’s what… Rabbi? Yes. Regarding a negative majority—how is that different from a majority not before us? What do you mean? I didn’t understand.
[Speaker E] A negative majority is a kind of majority not before us, because basically I don’t know the actual number of the probability.
[Rabbi Michael Abraham] I’m saying, usually a majority that is before us can be more of a positive majority than a majority not before us. Yes.
[Speaker E] But it’s not the same thing.
[Rabbi Michael Abraham] It’s not that one is negative and a majority that is before us is positive. It could be that I have many butcher shops in the country. I estimate that most of them are kosher. Fine? But I don’t have numerical data. Yet that is still a majority that is before us. Fine? It doesn’t have to be. You’re right that if I had to bet, a majority that is before us is more likely to be a positive majority than a majority not before us. So I want to continue. I now want to bring two additional examples that came up among the halakhic decisors as a consequence. And this too I heard from Rabbi Bas. I spoke with him before I wrote my columns on this topic, so it was following some conversation I had with him. So he says, for example, he cites Rabbi Yitzchak Elchanan, who wrote a permit based on two majorities regarding a ship that sank at sea. And he said that most of the passengers were gentiles, and most of those who drown die. So therefore there are two majorities, and one can permit the wives of those who sailed on that ship to remarry. Because there are two majorities here. Most are gentiles, so it’s not relevant at all, and the minority of Jews also fell into the sea, and most of them drowned. So, two majorities. Likewise, Rabbi Goren in several responsa relies on two majorities—for example, regarding the fallen of Kfar Etzion in the War of Independence. So he says: most of the Jews there in Kfar Etzion were bachelors, young guys; the minority were married, and most of them were killed. A few were taken captive, but most were killed. So, two majorities. That’s what he said. Since bachelors are not relevant, and among those who were married most were killed. Now here, in my humble opinion, this is a serious mistake. A really serious mistake. In Rabbi Goren’s case it’s two serious mistakes, and in Rabbi Yitzchak Elchanan’s case it’s one mistake. Again, I’m talking only about the two-majorities argument. It could be that there were other branches of argument there or other considerations that in the end made the permit justified. But this consideration of the two majorities that they brought is not correct. Why is it not correct? I’ll start with Rabbi Yitzchak Elchanan, because his mistake is also found in Rabbi Goren, except that Rabbi Goren has one more. With Rabbi Yitzchak Elchanan, when a woman comes before me asking permission to remarry, her husband was on the ship. Fine? And the ship sank. What are the two majorities he speaks of? Most there were gentiles anyway, the minority were Jews, and among the minority of Jews most drowned. So, majority of a majority. Fine? Therefore she is permitted to remarry. That is simply incorrect. Why? Because when the woman comes before me and asks—he was married, he wasn’t a bachelor. He was a Jew, sorry; not a bachelor, that’s already axiomatic. Meaning, he was a Jew, not a gentile. So what is the relevance of the fact that most were gentiles? Her husband was Jewish—that I know. Again, there is prior information here, right? Meaning, true, most there were gentiles, but here I have information. My information is that the person I am dealing with was Jewish. So what remains? What remains is only that most people who fall into the sea drown. That is one majority, not two majorities. You cannot permit her on that basis. When you talk about all the passengers on the ship, then you tell me: most were gentiles, and even among the Jews most drowned. But I’m not talking about all the passengers on the ship. What is before me—some woman who was married to all the passengers on the ship? It’s irrelevant. Unless you could make the following claim: suppose I knew that one person from that ship survived. There were a hundred on the ship, eighty gentiles and twenty Jews, one survived. Fine? Exactly. Now I ask: is that one who survived the husband of the woman before me? Suppose I don’t know. Here there may be room to speak about two majorities. Because I say: most were gentiles, so presumably the one who survived was a gentile, and even if he was a Jew, chances are he was not the husband of the woman. Well, then that’s—
[Speaker G] Really two majorities.
[Rabbi Michael Abraham] So here there are two majorities, but those two majorities are not relevant. Because I don’t know—if I knew one person survived and I asked whether this is the woman’s husband, that would be a question of two majorities. But that’s not the discussion. I don’t know who survived and who didn’t. I only know that a minority survived and there was a minority of Jews. But I am discussing a concrete person, the husband of the woman standing before me, and he is certainly Jewish. It is not relevant that most there were gentiles; he was Jewish. Regarding him there is only one majority: most who fall in drown, that’s all. There are no two majorities here.
[Speaker G] But for her to remain forbidden, it would have to be that some survived. If only a few survived—say one or two or however many, five, no matter—that’s clearly a minority, then everything the Rabbi just said holds, and the consideration really is two majorities.
[Rabbi Michael Abraham] No, but that’s not correct. I don’t know who survived and who didn’t, if anyone did at all.
[Speaker G] But if I know, say, that a minority survived.
[Rabbi Michael Abraham] There were a hundred people on the ship and ten survived. No—I don’t know, I have no idea. I’m not assuming anything, I have no idea. What I know is that most were gentiles and I know that most drowned. That’s all, that’s what I know. I have no idea; it could be that they all drowned. On average, a large majority drown in such a case. That’s it. You need to understand: this is an important lesson from some of the examples I brought earlier and from many others we’ve encountered throughout the series. How you ask the question is the most important thing in statistics. How do you ask the question? You have to ask the question very, very precisely. To understand what I know, what I don’t know, and what I want to know. Those are the three things I need to ask myself. And there are nuanced differences between very similar questions, and that’s because the formulation is imprecise and we’ll make a mistaken calculation. Usually, when the question is asked precisely—“the question of a wise man is half the answer.” The answer is already clear. You need to ask the question precisely. And here that is exactly the problem: they did not ask the question precisely. The question was not what is the probability that this person drowned—there you have two majorities. But here I am being asked by a concrete woman; her husband is a concrete person, I know who he is. Now I ask: did he drown or not? There is only one majority: he fell into the sea, and most likely he drowned.
[Speaker G] No, but if I know that most are gentiles… but if I know that only a minority survived, and that I do know. I know that only a minority survived.
[Rabbi Michael Abraham] If you know, for example, that one survived, and you ask whether that one is the woman’s husband, then yes indeed, then it’s two majorities. By the way, the two majorities there are the reverse—they’re not because… the two majorities there are that most drowned, meaning a minority survived; the minority are Jews, and even among the Jews only a minority survived. Fine? So how you ask the question is the most important thing in statistics. If you ask the question precisely, then you also know how to do the calculation and you arrive at the correct answer. In almost every case when you give an incorrect answer, it’s because you asked the question incorrectly. And there are very, very similar questions—the differences are really just nuances—like the questions I discussed earlier, like the quality of the judge versus the quality of the ruling. Right? Given that Reuven murdered, what is the probability that the judge will rule that Reuven murdered? That’s one question. Given that the judge ruled that Reuven murdered, what is the probability that he really murdered? A completely different question. The answers can differ radically. And it looks the same. After all, the judge is a high-quality judge, so if he said Reuven murdered, then obviously Reuven murdered, right? Because he’s a high-quality judge. Not related at all. Related, but very far from necessarily. Not necessarily at all. Or with contagious diseases—diseases, right? The question whether you have some rare disease—we discussed that too, right? So the quality of the test is like the judge; the quality of the test is ninety-nine percent. It misses only one percent of the results—it gets only one percent wrong. Now the test came out saying I’m sick. What’s the chance I’m really sick? If the disease is rare, the chance could be negligible, even though the test is ninety-nine percent reliable. Why? Because these are two opposite questions. The reliability of the test is the question: given that the person is sick, what is the probability that the test will indeed show it? That determines the reliability of the test. When I ask about the reliability of the result, I say the question is reversed. The test showed that I’m sick—what is the probability that in reality I really am sick? That is the reverse conditional probability, and it is entirely different. With rare diseases it’s just heaven and earth. The error rate is one percent and the accuracy of the test is ninety-nine percent, while the accuracy of the test result can be one percent.
[Speaker H] Because you’re basically looking at the entire population, most of which—
[Rabbi Michael Abraham] The overwhelming majority does not have it.
[Speaker H] We talked about that then, I won’t go back into the calculation—
[Rabbi Michael Abraham] But I’m saying: it matters enormously how you ask the question. And here is a wonderful example of that. Now I move to Rabbi Goren’s example, with the fallen of Kfar Etzion. Exactly the same thing. He says: most were bachelors, and among the married most died, so there are two majorities, therefore one can remarry. Right? That’s nonsense. Because the husband of the woman standing before me was not a bachelor. And I’m discussing the question whether he died, not how many of them died or how many married people died. I’m asking whether he died. So why do I care that most there were bachelors? There is only one majority there: most of the people died. So there is one majority that he died, but that’s one majority, not a majority-of-a-majority. This is a question about a specific person. Now here, as I said earlier, there is another mistake—
[Speaker I] Beyond what came out with Rabbi Yitzchak Elchanan.
[Rabbi Michael Abraham] What is the second mistake? Suppose the percentage of bachelors was, I don’t know, seventy percent. I don’t know how much; there were quite a few married people there. Say seventy percent bachelors. That majority is far from an overwhelming majority. Very far from an overwhelming majority.
[Speaker E] And how many died among the defenders of Kfar Etzion?
[Rabbi Michael Abraham] Also something like that. There were dozens who were taken captive. Again, not an overwhelming majority.
[Speaker A] So even if Rabbi—
[Rabbi Michael Abraham] Goren were right that it was a majority-of-a-majority—and he is not right about that—even if he were right that it was a majority-of-a-majority, a majority-of-a-majority still doesn’t help. It has to be the product of two overwhelming majorities in order to reach an absolute majority. Understand that basically there were several women here such that, if all of them had been permitted to remarry according to Rabbi Goren’s ruling, then several of them would have been married women who were wrongly permitted to remarry. The whole idea of two majorities is only in a place where in factual reality it is clear to me that the woman is permitted, that her husband is dead. My whole problem is only a formal problem. For it to be factually clear to me that the husband is dead, I need there to be an absolute majority here, or a product of two overwhelming majorities. That was not the case there. By the way, there, for example, it was really a majority that is before us. Michael. In Kfar Etzion it was a majority that is before us, right? There was a certain number of people there, of whom the majority—
[Speaker E] Basically you know, you can count the—
[Rabbi Michael Abraham] People who were there. In principle one could count them, even though we didn’t have the information. But we don’t have the information—though it’s still a majority that is before us. We really didn’t know in real time. Today we know, but we didn’t know in real time how many died; and how many were bachelors and how many were not, that we did know, I assume—meaning, they knew who the people were there. So that was even known. But how many were taken captive and how many died, that probably was not known in real time. In both cases it is a majority that is before us.
[Speaker G] So how did the Rabbi permit it? It really doesn’t make sense.
[Rabbi Michael Abraham] Right. He had mathematical knowledge, he studied, he knew—possibly. I haven’t read the responsum inside. I’m relying on what Rabbi Bas told me. It could be that in his responsum there are other aspects, that it was about someone for whom it was clear he had died, and he only needed to hang it on this mechanism of two majorities, because he was looking for precedents, right? To hang it on precedents. Fine, so I don’t know. So I think that if you are convinced the husband is dead, you don’t need to hang it on precedents; don’t bring me two majorities. Tell me there is an absolute majority that the husband is dead and the woman is permitted. The whole mechanism of two majorities is a formal mechanism; it’s not interesting. If you reached the conclusion that this is an absolute majority, then permit her to remarry even without hanging it on the fact that it is two majorities. Like when someone sees the identifiable face with the nose—that’s the example I gave, right? So when a witness comes and says, “I saw the identifiable face, I recognize the person,” there is some chance he was mistaken. But it is convincing enough to me to count as an absolute majority. An absolute majority is not certainty, but it is an absolute majority. I know the husband is dead. In such a case I permit her to remarry. What is this, two majorities? There are no two majorities there. It is one majority, not two majorities. But it is an absolute majority. If it is an absolute majority, you do not need a product of majorities. Water with no visible boundary is an overwhelming majority, not an absolute majority. And therefore you need a product with another majority. Now in Rabbi Goren’s case it wasn’t even an overwhelming majority; it was just an ordinary majority. The product of two ordinary majorities—that would be outrageous to permit on that basis. Suppose there were a hundred, I don’t know, a hundred people there, of whom twenty or thirty were married; he permitted thirty women to remarry, and among those thirty husbands, say twenty or thirty percent had been taken captive. Nine women who were still married women—he permitted them to remarry. I say, again, I’m not slandering him or anything; I haven’t read the responsum inside. It could be he had other supporting considerations and it was fine. I’m only saying that as far as this argument itself goes, this two-majorities argument doesn’t even get off the ground.
[Speaker A] It really doesn’t get off the ground. But in Kfar Etzion, actually, they didn’t go into captivity there; only four remained from the whole settlement.
[Rabbi Michael Abraham] In captivity—what do you mean? They were taken captive in Jordan. They were massacred, no? No, some were massacred, and some were taken captive to Jordan.
[Speaker L] A relative of mine was a prisoner there and was released.
[Rabbi Michael Abraham] Yes, there were many; I know people too.
[Speaker A] But I mean, there I remember, with this whole story when it happened, I remember reading that it was ninety-something percent of them who were massacred.
[Rabbi Michael Abraham] Doesn’t matter—even ninety-something percent still leaves married women whom he permitted to remarry.
[Speaker A] No, the question is whether according to this approach that would be an absolute majority or an overwhelming majority.
[Rabbi Michael Abraham] No, it’s not an absolute majority. It’s barely an overwhelming majority, if that. Ninety percent, in my opinion, isn’t even an overwhelming majority. Right. Listen, ninety percent means that there is—
[Speaker A] Ten percent that her husband is alive.
[Rabbi Michael Abraham] You would permit ten women to remarry when one of them is still married? That is completely unreasonable.
[Speaker A] What is Rabbi Goren’s answer to that? What did he say?
[Rabbi Michael Abraham] I said, I don’t know. I heard about this from Rabbi Bas, so as I said, this is not criticism of Rabbi Goren; it is criticism of this particular argument that I heard from Rabbi Bas. I don’t know what is going on in Rabbi Goren’s—
[Speaker F] Rabbi Goren—I haven’t read the responsum. No, it could be he brought something else in addition.
[Rabbi Michael Abraham] I said, it could be, it could be. That’s why I’m not criticizing him; I’m criticizing the argument. Okay? I don’t know; I haven’t read it inside. So, uh… Rabbi, Rabbi, sorry—in the first case, in the first case, where you don’t accept at all, even before getting into overwhelming or absolute majority, you don’t accept taking into account the majority that most were gentiles on the ship that sank. Because you say the question is about the specific person, whom we know is Jewish. Correct. But—but since we don’t know at all what happened to this person, what happened to any of them, then why not take the majority into account in order to permit? Because overall I know he is Jewish, but I don’t know where to place him, among those who died or not.
[Rabbi Michael Abraham] So he—no, there is one majority, you’re right. But it’s only one majority, not two.
[Speaker F] And also that most drowned. But because I have no information whatsoever on this person, on this Jew I’m asking about, whom his wife is asking about—
[Rabbi Michael Abraham] What information do you want? You know he is her husband and you know there’s a majority that he drowned. So that is one majority. Why isn’t it two majorities?
[Speaker F] No, but I also use the fact that there was a minority of Jews there—
[Rabbi Michael Abraham] And many gentiles, so what?
[Speaker F] No, because basically it starts differently: I use the minority of Jews who were there and also the minority—
[Rabbi Michael Abraham] But why? When you discuss the question of what happened to him, there is no majority that he is a gentile and a minority that he is a Jew; he is certainly Jewish.
[Speaker I] Why do I care that there were gentiles around him?
[Rabbi Michael Abraham] I am discussing him. That’s exactly the point. We are talking here about conditional probability. This is about a particular person, and about him I ask the questions. It is not an a priori question of what happened to the typical person on this ship. I’m asking a question about a specific person.
[Speaker M] And given that on… yes.
[Rabbi Michael Abraham] The woman standing before me is a particular woman, and she has a known husband. Now this is a bit like those balls, right, from among all the balls. When I want to draw the black ball, even if it is one ball and there are ninety-nine white ones, I will draw the black one. Why do I care that there are ninety-nine white ones around it? It matters enormously how you ask the question, what the data are, and what I want to know. What do I know, what don’t I know, and what do I want to know. Those are basically the three key questions: what do I know, what don’t I know, and what do I want to know. That is what determines the precise formulation of the question. If we determine that correctly, we will usually do the calculation correctly. The confusions always come when we don’t determine correctly—many times we don’t correctly determine what it is we want to know. Okay, we’ll stop here.
[Speaker J] But one last point—in the end this basically means that what matters to you in the end is the number, like how many percent it really is so.
[Rabbi Michael Abraham] That will be the discussion at the beginning of the next class. Is the permitting of agunot based on statistics? Shmuel asked that in the previous class; I said we’d get to it, and we’ll get to it next class—I just didn’t get there this time. Okay, comments or questions?
[Speaker E] Thank you very much, thank you very much, thank you very much.
[Rabbi Michael Abraham] Okay, thank you, Sabbath peace.
[Speaker I] Can I just get the link to the video with the math, with that probability?
[Rabbi Michael Abraham] I’ll upload it to the group, to the thread. Excellent, excellent, thank you.
[Speaker I] Goodbye.