Doubt and Probability—in Halakha, Thought, and in General—Lesson 49—Rabbi Michael Abraham

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Opening: review of the conditions for leniency in a double doubt — the two requirements commonly presented by halakhic decisors were introduced: that it not be “all one kind of coercion” and that it be a reversible double doubt.
The distinction between the two conditions — it was emphasized that there is no full identity between “one category” and “reversible,” and directions were given in which one exists without the other.
Negative doubts versus positive doubts — it was explained that the whole discussion of a double doubt belongs specifically to negative doubt, where we do not have an objective probabilistic weight but rather a lack of knowledge.
Why we need a double doubt at all — since real probabilities cannot be calculated in a negative doubt, Jewish law uses a formal counting of sides in order to maximize the chance of leniency.
The meaning of “all one kind of coercion” — distinct grounds for leniency are counted, not every factual variation separately; several forms of coercion are considered one halakhic branch.
The need to explain the requirement of reversibility — since this is not necessarily the same requirement as “one category,” it needs an independent explanation why a double doubt must be reversible.
A probabilistic introduction: dependence and independence between events — through examples of rolling dice, it was explained when probabilities are multiplied and when one must move to conditional probability.
Conditional probability as a model of thought — it was clarified that with dependent events, one calculates the likelihood of B given A, not the ordinary absolute likelihood of B.
Clarification through students’ questions — the die roll was compared to the “two children” question, to show that statistical dependence comes from relevant information and not from physical influence.
The Monty Hall paradox and Bayesian inference — the example was used to illustrate a change in probability בעקבות new information, and the idea of learning from experience.
Additional examples of probabilistic mistakes — crib death, an artillery crater, and regression to the mean in basketball were discussed as illustrations of confusion between dependence and independence.
Application to a reversible double doubt — it was suggested that in the reversible case there is no dependence between the doubts, and therefore they can be treated as two events whose product reduces the side of prohibition.
The limitation of the explanation: the case of an open entrance — it was clarified that a double doubt that is not reversible is not necessarily statistically dependent; sometimes the lack of reversibility is only a content-linguistic problem.
A moderate conclusion about the requirement of reversibility — the rule that one is not lenient without reversibility was explained as a halakhic “no distinctions” rule: out of concern that sometimes lack of reversibility does reflect real dependence.
Narrowing the sample space and its applications — examples were presented involving legal presumption, “what is under a person’s hand is his,” and a representative sample in science, as illustrations of conditional probability.

Summary

General Overview

The lecture continued the analysis of the laws of a double doubt, especially the two common requirements for being lenient with it in a Torah-level case: that it not involve “one category” (for example, several forms of coercion that all count as one ground for leniency), and that it be a reversible double doubt — one where the two doubts can be formulated in either order. Rabbi Michael Abraham wanted to clarify whether these two requirements stem from the same logic, and he offered a broad probabilistic-conceptual analysis in order to understand their basis.

## A double doubt belongs to negative doubts
The central assumption is that the whole discussion of a double doubt does not deal with “positive” doubts, where we have known and calculable probabilities, but with negative doubts — situations of lack of knowledge. If we knew the probabilities, we would simply check whether there is a majority in favor of leniency. Therefore, the laws of doubt and double doubt are an alternative halakhic mechanism, meant to make action possible in a world of partial ignorance. In a double doubt there is no certainty that there is in fact a majority for leniency, but there is a halakhic attempt to increase the chance that there is.

## Why “all one kind of coercion” makes sense
Since in a negative doubt one cannot really weigh probabilities, Jewish law “counts sides.” But counting sides is itself a halakhic determination, not an objective statistical one. Therefore, if three different branches all lead to the same basic ground for leniency — for example, three forms of coercion — they are not counted as three separate sides but as one side. That is the logic of “all one kind of coercion”: not every sub-possibility counts as an independent doubt, only a distinct ground does.

## Is reversibility connected to dependence between events?
To explain the requirement of “reversible,” Rabbi Michael Abraham moved to a discussion of probability: when two events are independent, their probabilities can be multiplied; when they are dependent, one must use conditional probability. Examples involving dice rolls, information about an odd-number result, and a loaded die illustrated how earlier information changes the assessment of the next event.

From here a possible explanation was suggested: maybe a reversible double doubt means that the two doubts are independent, and therefore one can “multiply” their lenient force; whereas if it is not reversible, there may be dependence, and therefore the force of leniency is weakened.

## Monty Hall, Bayes, and statistical dependence
Through the Monty Hall paradox, Rabbi Michael Abraham clarified that new information can narrow the space of possibilities and thereby change probabilities. This is the idea of conditional probability and Bayesian thinking: the question is not only what the absolute chance is, but what the chance is given relevant information. Examples such as two crib deaths, an artillery crater, and three-point shooting in basketball were used to show how much people get confused between dependent and independent events.

## The limitation: lack of reversibility is not proof of dependence
At this point Rabbi Michael Abraham returned to the example of an open entrance: there is doubt whether the husband is expert at detecting an open entrance, and even if he is expert, there is doubt whether the intercourse was by coercion or consent. Here the doubt is not reversible, but not because the events are statistically dependent; rather, because in the reversed formulation a content problem is created: once you have already assumed there was an opening, you can no longer then doubt whether there was an opening at all. In other words, lack of reversibility can stem from the structure of language and content, not from real dependence between the events.

From this comes the conclusion: the probabilistic explanation for the requirement of reversibility is only partial. It is true that when there is real dependence there generally will not be reversibility, but not every lack of reversibility indicates dependence.

## Conclusion: a halakhic rule of “no distinctions”
Therefore Rabbi Michael Abraham suggested understanding the requirement of reversibility as a sweeping halakhic rule: since sometimes lack of reversibility really does express dependence that weakens the double doubt, the halakhic decisors established a rule that one is not lenient with a double doubt that is not reversible, even if in some cases the reason is not dependence but the structure of the content. This may also explain why some later authorities do not accept the requirement in an absolute way.

## Narrowing the sample space: further applications
Toward the end, Rabbi Michael Abraham returned to the idea that conditional probability is basically a narrowing of the sample space. He applied this to the presumption that “what is under a person’s hand is his”: one cannot infer from the general majority of objects in the world to the case of objects involved in legal dispute, because that is a subgroup that may not be a representative sample. From here also comes the connection to scientific generalizations: any generalization is valid only if the observed sample represents the whole.

The lecture ended with the announcement that next time two more Talmudic examples would be brought of narrowing a majority by means of conditional probability, and after that the responsum of Havot Yair that would conclude the series.

Full Transcript

Okay. Good evening. Good evening.

[Rabbi Michael Abraham] We were in the topic of double doubt. We saw that there are two requirements commonly accepted by the halakhic decisors in order to apply this idea that a double doubt is treated leniently. Meaning: an ordinary Torah-level doubt is treated stringently, but in a double doubt, a double uncertainty, then one usually goes leniently. In rabbinic cases there’s no practical difference whether it’s a single doubt or a double doubt, because both are treated leniently. So why, and when, do we go leniently in a double doubt even in a Torah-level case? Two things have to exist. I said that at least one of them is disputed, but in principle it is common to require two conditions. First, that it not be what’s called “one category of doubt, one category of coercion,” meaning that the different sides of the doubt should not really be one side, but rather three against one, in the tree structure I described. That is one category of doubt… and one category of coercion, sorry. And also, that it be a reversible double doubt. Meaning, you should be able to formulate doubt A first, and if you conclude that it is prohibited on the basis of doubt A, there is still room to permit it on the basis of doubt B. And you should also be able to formulate it as doubt B, and if you conclude that doubt B leads to prohibition, there is still room to permit it on the basis of doubt A. In other words, the order in which the doubts are presented should not matter. That is called a reversible double doubt. If it does matter — meaning, if the doubts can only be presented in one order and not in the reverse order — then we do not go leniently on the basis of double doubt. I said there is a dispute about this, but it is commonly accepted among the halakhic decisors that this is a requirement in order for us to be lenient in a double doubt. I said that quite a few later authorities tend to identify these two things, and we saw examples with the slaughtering knife and so on, where the fact that the double doubt is not reversible and the fact that it is all one category of doubt are really the same thing. That really does often appear in the same form, but last time I showed that it’s not actually the same thing. In other words, there is a double doubt that is from one category and yet is reversible, and there is a double doubt that is not reversible and yet is not from one category. I brought examples in both directions. And why is that important? Because when I have — that is, the requirement that it not all be one category of coercion has a simple explanation. I said that we are dealing with negative doubts and not positive doubts, because in positive doubts the whole law of double doubt is irrelevant. We simply look at the chance of leniency and the chance of prohibition; if there is more than fifty percent chance for leniency, then we permit it. And therefore it really makes no difference whether the calculation that gets me to those fifty-five percent or those eighty percent comes from multiplying probabilities or from one direct calculation. At the end of the day, if there is a majority, there is a majority. The entire law of single doubt and double doubt was said only regarding negative doubts. Because in negative doubts we do not have any calculation that can finally tell us whether there is a majority here or not. And therefore the way Jewish law ensures that even though the doubt is negative, we still maximize the chances that there really is a majority for leniency, is to require a double doubt. Okay? Now, once these are negative doubts, then in fact we are not talking here about a fifty-fifty doubt. We are talking about two possibilities where I have no knowledge, no way to weigh each one and see which has a greater likelihood. This is absence of knowledge, right? A negative doubt is a doubt arising from ignorance. A positive doubt is a doubt arising from knowledge, where I know it is fifty-fifty. A negative doubt is a doubt arising from ignorance. I know nothing, so I assume fifty-fifty, but that is only an assumption that comes from ignorance. In such a situation, even if there is a double doubt, in the end it may be that in fact there is no majority for leniency. Because if it were positive fifty-fifty, then yes, there would be a seventy-five percent majority for leniency. But if it is negative, then it could definitely be that even though three sides point to leniency and one side points to prohibition, still most likely it is prohibited. It depends on how much weight each side carries. Since I do not know, I assume they are equal — but that is only because I do not know. So in that situation we need to find some other way to function in a world where there are doubts. Because if we prohibit all doubtful cases, even in negative doubts we would prohibit all doubtful cases, because we would always assume stringently that maybe there is a majority for stringency. We would never get anywhere. Because almost all the doubts we have in the world are negative doubts, and we would be unable to move in the world. So therefore the Sages say: let us require a case of double doubt. If there is a case of double doubt, we increase the chances that in the bottom line there really is a majority for leniency. Not guarantee a majority for leniency — it could still be that there is in fact a majority for prohibition — but it improves the situation more than a single doubt does. That is our way to ensure, or not ensure but improve the likelihood, in the absence of knowledge, that there is nevertheless a majority for leniency here. Kind of like what we saw with two majorities, which was also a negative majority, and there too there was basically a formal requirement meant to offset the fact that we were dealing with a negative majority. Why is this important? Because if that is the case, then what we really need to do here is count sides rather than probabilities. And if there is a majority of sides, then we go toward leniency: three sides against one. But the definition of sides — we saw in previous sessions — is arbitrary. You can always add all sorts of sides one way or another. Yes? Doubt whether it’s pork or beef. But maybe the pig is an Indonesian pig, maybe it’s a warthog, maybe it’s this kind of pig — so I have many sides toward prohibition, and only “beef” as one side toward leniency. If you want, enlarge the number of sides on the beef side too. You won’t get anywhere. In the end, we have no way to define objectively what counts as a side or how one counts sides. And since that’s so, we are supposed to decide on some halakhic rules that have some sort of logic behind them, but it is certainly not something absolute, objective, statistical. We have to decide what counts as a distinct side, a separate side. And that decision is a halakhic decision, not a probabilistic one. And where it is all one kind of coercion, we do not see that as several sides but as one side. So when we count sides now, there really will not be a majority of sides toward leniency. Because seduction of a minor is a different type of coercion than coercion of an adult woman. But it is still coercion. Meaning, the clause of leniency is because coercion occurred here. So there can be coercion with a gun, there can be coercion by force, there can be all kinds of things, and there can be seduction of a minor. That too is a kind of coercion. And all of these are really the side of coercion. Therefore it is one side. And therefore, if the three lenient sides are all forms of coercion, then even if there are three sides for leniency, from my perspective it is one side. It is all one kind of coercion. Since the counting of sides is halakhically arbitrary anyway, Jewish law is also what determines what is called a side. Therefore, if it is all one kind of coercion, clearly we cannot count different kinds of coercion as several sides. One second, let’s silence this lady here, one moment. Right — dogs bark and the caravan moves on. Fine. So the claim is that “all one kind of coercion” is a sensible principle. Not statistical, but sensible. Okay? Given the fact that we are dealing with negative doubts, the way we ensure — or maximize — the possibility that there really is a majority for leniency is by requiring two doubts, a double doubt. Two doubts ensure that there is a majority of sides for leniency, but in order for there to be a majority of sides for leniency I have to count how many sides there are for prohibition and leniency. How do you count sides? You have to decide that each ground of leniency counts as a side. So if one ground of leniency, namely coercion, can appear in three different branches of the double doubt, from my perspective that is one category of coercion, one branch. So there is one branch for leniency and one branch for prohibition; we are in a balanced doubtful situation. Therefore, “all one kind of coercion” follows directly from the logic of double doubt in negative doubts. Of double doubt, period. Because double doubt is always in negative doubts. So that is regarding “one category of doubt,” or “one category of coercion,” sorry. Now, if I say that a reversible double doubt is the same thing as “all one kind of coercion,” then I do not need a separate explanation for reversible double doubt. Right? The same explanation, because it is the same principle. But if, as I said last time, that is not correct — meaning, if the requirement that it be reversible and the requirement that coercion not all be one category are not the same requirement — then I find myself needing to explain this rule that a double doubt must be reversible. Because I cannot use the explanation of “one category of doubt” or “one category of coercion,” since it is not the same principle. If it is not the same principle, then it needs an explanation of its own. So last time I brought one explanation and said it was questionable. The second explanation — which I only said in general terms, just to close the loop — is that the requirement that the doubt be reversible comes from the fact that I want the probabilities of these two doubts to multiply one another. And here I… maybe I’ll give some kind of introduction. When we talk about complex events, how do I estimate the likelihood of complex events? That depends on the relationship between the events. If the two events are independent, then the chance of both events occurring is the product of the probabilities. Say I roll two dice. What is the chance that both show three? Okay? The chance is one-sixth times one-sixth. The chance that the first die shows three is one-sixth; the chance that the second die shows three is also one-sixth. The two events are independent, so the chance is one-sixth times one-sixth, which is one out of thirty-six. But all that is only because the events are independent. But if the events were dependent — for example, if the die is not fair and I roll it twice — then if the first time it lands on three, assuming the first time it landed on three, the chance that the second time it also lands on three is no longer one-sixth; it is much higher. Why? Because apparently this die has some tendency to land דווקא on three. And therefore, if once it came out three and I know it is not fair, then the chance that the second time it will also come out three depends on the first event. The second event depends on the first; there is dependence between the events. And once there is dependence between the events, then in fact I cannot multiply the probabilities. I cannot say that the chance of getting three on both is one-sixth times one-sixth. Since I do not know the bias of the die, then for the first result I will guess that it is one-sixth. The chance of getting three is one-sixth, because I don’t know — negative doubt, right? I know the die is not fair, but I don’t know in favor of which face it is unfair. So I say: under that assumption, the chance that it is biased in favor of three is one-sixth. Fine? So if you ask me, what is the chance that the first roll will be a three, I’ll say one-sixth. But after it has already come out three, then there is a very high chance that apparently the bias of the die is in favor of the face three, because I know that three came out. Since that is so, on the second roll I will no longer assume that the chance of getting another three is one-sixth. Here the chance is already much greater — one-half, I don’t know, five-sixths, however you estimate it — but if I already have reason to assume that the die is biased toward three, then on my second guess, if you ask me what the chance is that it will again come out three, obviously it’s not one-sixth, right? It’s more. That means the events are dependent on each other. Right? The second event depends on the first: if I know the first event, the probability of the second event changes. Now notice that the way I formulated it just now is a slightly different formulation of the same idea. Dependence between events in statistics, in probability, is often described through conditional probability. What does that mean? Suppose I ask what the chance is that three will come out. Let’s say the event that three comes out we’ll denote as A. What is the chance of A? P of A, the probability of A, the chance of A — one-sixth. Okay? Now they ask me what is the chance of B? Meaning, that on the second roll as well it comes out three. Also one-sixth. What is the chance of A and B? P of A times P of B — one out of thirty-six. P of A and B is P of A times P of B. Okay? What happens when there is dependence? When there is dependence, then let’s go back to the example I gave before: if the first die came out three, then P of A is one-sixth because I do not know in which direction the bias goes, so the chance of getting three is one-sixth. But now I ask: what is the chance that the second roll will also be three, given that I know the first roll was three? That chance is no longer one-sixth, right? I really have to take into account here not P of B, the chance of getting three, but the chance of getting three given the fact that on the previous roll it came out three. In other words, I multiply P of A not by P of B, but by P of B given A. Suppose if—

[Speaker C] I want to know what—

[Rabbi Michael Abraham] —the chance is of getting three twice, then it’s the probability of getting a three the first time times the probability of getting a three the second time given that the first time was a three. Okay? That product gives me the combined probability of getting three in both of those cases. Now if the events are independent, then P of B given A — meaning, the chance of getting three on the second roll given that you got three on the first roll — is the same chance as just getting three, right? P of B given A and P of B are the same thing, because the fact that A happened changes nothing. That’s what it means to say there is no dependence between A and B. In other words, the mathematical way to express lack of dependence is basically to say that P of B given A equals P of B. Meaning, the fact that A happened does not change the probabilities. Okay, suppose I roll a die and ask what is the chance of getting five. Let’s say A is “it came out five.” P of A is one-sixth. Okay, now I say yes, but it is known that an odd number came out. It is known that an odd number came out. Now they ask me: what is the chance that it was five? Here it is no longer just P of B, it is P of B given A. What is the chance that it was five given A, that an odd number came out? And now the result is one-third, not one-sixth, right? The odd numbers are one or three or five. So the chance that it was five is one-third. Okay? So that means that the information I gave you, that A occurred, changes the chance of B occurring. Meaning, P of B given A is not the same thing as P of B without A being given. P of B given A is not equal to P of B. That means there is dependence between A and B. If there is no dependence—

[Speaker C] between—

[Rabbi Michael Abraham] —A and B, say: what is the chance of getting five? I say one-sixth. Now I know that the die is made of metal. That is A. And now I ask what is the chance of getting five given that the die is made of metal? Still one-sixth. It doesn’t matter whether the die is made of metal or wood as long as it’s fair. Right? So that means the events A and B are not dependent, because P of B given A is the same thing as P of B. The fact that A happened does not change the odds, it’s uninteresting, irrelevant. Therefore that is the mathematical expression of the fact that A and B are independent events.

[Speaker D] Wait, if I say that on the first roll it came out five, that won’t affect the second roll?

[Rabbi Michael Abraham] If the die is fair, then no.

[Speaker D] So then why in that puzzle the rabbi brought us with the children, with the two children? Because there they didn’t say it was the first one.

[Rabbi Michael Abraham] Of course — on the contrary. There the whole idea is that maybe there is dependence and maybe it’s some genetic problem or something like that because of which the first child died, and that’s also the reason the second one would die?

[Speaker D] No, no, not that example. The puzzle the rabbi gave us, where you tortured us a few weeks ago, with the woman who says “I have one boy,” she has two children and we know that one child is a boy, and the question is what is the chance that the other child is a boy or a girl?

[Rabbi Michael Abraham] Who was born on Tuesday, yes.

[Speaker D] No — what I’m saying is, even though there’s no biological dependence there. There’s statistical dependence, not biological.

[Rabbi Michael Abraham] Meaning, the information I gave you changes the calculation. So that means there is statistical dependence between the events.

[Speaker D] So why here with the die doesn’t it change? Because if I were to say—

[Rabbi Michael Abraham] It doesn’t matter whether the die is made of metal or wood.

[Speaker D] No, not wood. I’m talking about if I know that one of them came out five. He rolled the die twice. One time it came out five.

[Rabbi Michael Abraham] I’m asking, what is the chance that—

[Speaker D] The second time it also came out five?

[Rabbi Michael Abraham] Fair. Completely fair.

[Speaker D] I know that it’s fair.

[Rabbi Michael Abraham] So then what difference does it make that it came out five? So how is that different from the children?

[Speaker D] No, because with the children there is dependence.

[Rabbi Michael Abraham] I don’t understand the question. With the children those are events that have dependence. What dependence?

[Speaker D] If you know one of the children is a boy and he was born on Tuesday, that changes the calculation of the probability. Even though biologically it has no effect at all.

[Rabbi Michael Abraham] Statistically, the fact that he was born on Tuesday matters. It affects my information.

[Speaker D] So why doesn’t the die affect it?

[Rabbi Michael Abraham] What’s the difference between a child and a die? It doesn’t affect reality in the world. It affects my information. That information is relevant information. It changes the calculation of the possibilities now before me. It’s like knowing that an odd number just came up on the roll. That doesn’t cause the probability of five to increase now. But statistically, yes. There is statistical dependence, purely mathematical-conceptual.

[Speaker D] Why is it that when I rolled a die twice, one time it came out five, and I ask what the chance is — and it’s completely fair — what is the chance for the second roll? The rabbi says there’s no connection. But when two children were born, which is also some kind of random fifty-percent situation, and the first time a boy was born, then the rabbi says the second time it’s no longer fifty percent because there is dependence between the cases. What’s the difference between a die and a child? Why?

[Rabbi Michael Abraham] Because there we did the calculation. I showed you that the calculation is dependent. What do you mean?

[Speaker D] So why is the calculation here not right?

[Rabbi Michael Abraham] Because do the calculation and you’ll see that there is no dependence. Do the same calculation and you’ll see that you get nothing. You still get one-sixth. Nothing changes. Fine, we won’t go into that again. We did that calculation there. I’m saying: the point is that when the information I receive is relevant to the problem, that means that now, with the new information I have, the calculation can give a different result. And therefore from my perspective there is dependence between the events. Dependence between events means that information about the occurrence of one event affects the calculation regarding the likelihood of the second event. When there is no dependence, that means that this information is not relevant to that calculation and will not affect it. So the question whether the die is made of metal or wood will not affect in any way the calculation of the second probability. Therefore P of B given A — yes, the chance of getting five given that the die is made of metal — is just the chance that the die will come up five. P of B given A equals P of B. That means that being told A did not change the calculation, because A and B have no dependence between them. But if I am told that an odd result came up — that is A — then that does change the calculation. It changes the calculation because now it is one-third and not one-sixth. So that means P of B given A is not the same as P of B. P of B is one-sixth, the chance of getting five. But P of B given A, the chance of getting five given that an odd number came up, is one-third. So that means these events are dependent, statistically dependent. Now, so that basically means—

[Speaker C] Rabbi, can I say something? Yes. I think with the metal die example, what’s different here is not that there is or isn’t dependence, but simply that the information that the die is metal is not relevant. Right. It’s not relevant to the calculation. It’s not a matter of whether I do multiplication or whether I calculate this given that. That’s exactly it.

[Rabbi Michael Abraham] That’s what dependence is: it’s not relevant, therefore there is no dependence.

[Speaker C] No, I mean that I’m not moving to a different form of calculation of P of B given A. You’re not moving to a different form of calculation because there is no dependence. No, the probability of that is one, the probability that the die is one, and still there is multiplication. The probability that the die is metal is one.

[Rabbi Michael Abraham] No, it’s not one.

[Speaker C] They tell me, it is given to me that the die… No, I don’t know.

[Rabbi Michael Abraham] I have dice scattered around, with a distribution of metal dice and wooden dice, and I randomly take a die from the jar. Fine? I don’t know whether it’s metal or wood, but all of them are fair. Now I roll the die. What is the chance of getting five? It does not depend on the event of whether it turned out to be wood or metal. Right, okay. So dependence between events is expressed by whether the conditional probability equals the absolute probability; if they are equal, that means there is no dependence. If there is dependence, then the conditional probability differs from the absolute probability. Okay? There is — if you remember — we spoke once at the beginning of the series about the Monty Hall problem. There used to be a television game show where the host, the contestant — or player — stands in front of three closed doors. Behind one of them there is a goat, behind another there is a Mercedes, and the third is empty. Okay? Now he has to open a door, and he gets whatever is behind it. His goal, of course, is to open the door behind which there is a Mercedes. Okay? Now he chooses a door — say doors A, B, and C. Let’s say he chose door C. Fine? Now the host says to him, “Look, I’m opening one of the doors.” Fine? He opened, say, door A, and it turned out to be empty — there was nothing behind it. So now he has before him two doors, B and C. Earlier he had chosen C; now before him are doors B and C. The host says to him: “You can change your choice. You can stay with the door you chose originally, stay with C, or you can switch — you can choose…” He opened B, I don’t even remember what I said. Let’s say he opened B. Now you can switch to door A or stay with the door C that you originally chose. “What do you prefer?” At first glance everyone says, “What difference does it make? The probability is one-half.” Meaning, either the Mercedes is behind door C or behind door A. Two doors, so the probability is one-half. So what difference does it make whether he switches or doesn’t switch? It changes nothing. But the answer, of course, is that it changes a lot, and he should switch.

[Speaker D] But only on condition that the host knows what’s behind—

[Rabbi Michael Abraham] Yes, yes, the host knows. The host knows, of course, and intentionally opens an empty door. Right, that’s obvious. Now the point is: why should he switch? On the face of it, what difference does it make? There are two doors; the odds are the same for both; what difference does it make whether he stays with C or switches to A? The answer is that he has now received further relevant information. And the probability he now has to apply is conditional probability, not absolute probability. The absolute probability for door C is one-third, right? At the beginning there were three closed doors. What is the chance that the Mercedes is behind door C? One-third. Okay? The chance that the Mercedes is behind the other two doors is two-thirds. One of the other two doors — that’s two-thirds, right? Now the host opens one… one of the doors, but he always opens the empty door. That is his policy; he always opens the empty door, right? So what comes out is: what is the chance that the Mercedes is behind door A, the other closed door? Two-thirds. Not one-half. Why two-thirds? Because the chance for door C was one-third. The chance for one of the other two doors was two-thirds. Now the host opened one of those doors, and it was the empty one. Right? But no matter where the Mercedes is, he will always open the other one — the empty one. Therefore the door that remains closed has a two-thirds chance of having the Mercedes behind it. And therefore it is always worthwhile for you to switch, because you raise your odds from one-third to two-thirds. Now this is not intuitive; people get very tangled up with this, they call it a paradox, but it’s not a paradox. It’s just a simple statistical result. It’s just a little confusing intuitively, but there’s no paradox here. The answer, the calculation, is two-thirds. So now, what’s the point? What lies behind this? What lies behind it is that people do not understand that the information that the host opened one of the doors and it was empty is relevant information to the problem. So what does that mean? That the chance that the Mercedes is behind door A now, given that the host opened door B, has changed. At first it was one-third, but after I received information it rose to two-thirds. Because now it is a conditional probability. The chance that the Mercedes is behind door A given that the host opened door B is not the same chance as the chance that the Mercedes is behind door A in the absolute sense. The conditional probability differs from the absolute probability, and that means there is dependence between the events. The information that the host opened door B is relevant information to the problem. Therefore the probability calculation now changes after that event occurred. So really the problem is that we are applying absolute probability, whereas in truth we need to apply conditional probability. The events are dependent.

[Speaker F] Why, after he opened door A and it was empty, why exactly is it not one-half? After all, now there are only two doors left.

[Rabbi Michael Abraham] No, but he always opens the empty door. He doesn’t open a door at random and then, oops, it happened to be empty. If that were the case, you’d be right. But no. He has a policy: of the two other doors, he will always open the empty one. That is the policy. So he says now that the chance that the Mercedes is behind one of those two is two-thirds, right? And the host will always open the empty one and leave the Mercedes behind the closed one, assuming the Mercedes was behind one of those two. So that two-thirds remains.

[Speaker F] I still don’t really understand, because I’m saying: the person who is now going to choose, after he already knows that one door is empty, is now basically standing between two doors. It’s completely clear that one of those two has the car. So now he is in a new situation where that original one-third doesn’t matter, because now he has two choices.

[Rabbi Michael Abraham] Up to the last sentence I agreed. It’s true: there are two doors and behind one of them is the car. But that it doesn’t matter — that’s not true. Because the fact that the host opened door B is relevant information. Once I know that, the calculation of the chance behind door A changes.

[Speaker F] The calculation now is… I agree with what the rabbi is saying, that the empty door changed the odds. But it didn’t change them — what I don’t understand is why it changes them to two-thirds and not one-half. You now know that one of the three doors is empty.

[Rabbi Michael Abraham] Ah, you’re saying it should change to half-and-half between the two? Yes, yes. Because the chance of door C, the one you originally chose, was one-third, right? Opening door B changes absolutely nothing about the chance of C, because the host will never open מראש — his policy is to open either A or B; he has no policy of opening C. So the fact that he didn’t open C gave you no relevant information about door C, because he wasn’t going to open C in the first place. But the fact that he didn’t relate to door C changes nothing in your assessment of door C, because he wasn’t going to open it anyway. By contrast, regarding doors B and A, he has a policy that says: of those two, I will always open the empty one. So that means you discover something new if he opened B. If he opened B, that tells you something about what is going on with door A. It is relevant to the odds of door A; it is not relevant to the odds of door C.

[Speaker C] Rabbi, there’s another way to explain why this happens. If you look at it in reverse, you say: when I choose while there are three doors, I choose one of them, that means I have a two-thirds chance of being wrong. Now once he opens the second door for me, that means that basically the two-thirds moves to the third door, the one I didn’t choose.

[Rabbi Michael Abraham] Not the second door — that’s what I explained.

[Speaker C] Right. So basically all the probability funnels into the door that remained closed.

[Rabbi Michael Abraham] Yes. Exactly. In other words, the host basically makes the probability of the two other doors collapse onto door A. Because the host will always open the other one, so that changes nothing. It still remains two-thirds on those two doors when the host opens the empty door in advance. He knows. So it stays two-thirds. It makes no difference. Okay. In any case, this Monty Hall problem is just another example of the confusion we have regarding conditional probabilities. And very often people talk about Bayesian thinking. Because Bayesian thinking, as it’s called in probability or statistics, is basically thinking that learns from experience. Meaning, if things happened, I draw conclusions from them and my information improves. Okay? That’s basically what happens when we learn from experience. Things happen to me, I process that information, and now for future cases I come in wiser. Fine? That’s called Bayesian inference. In other words, Bayes’s theorem deals with conditional probabilities, never mind, but Bayesian inference is basically learning from experience. And when I talk about the difference between probability and conditional probability, that’s simply a reflection of the fact that experience teaches me things. Meaning, as a result of experience I become wiser. I know how to do the probabilistic calculation better if the experience is relevant. But if I have experience in irrelevant areas, then the fact that I have experience changes nothing. For example, suppose someone has a great deal of experience in physics. Fine? Now the question is whether he’ll succeed better at analyzing poems. Probably not so much. Why? Because your experience in physics isn’t relevant to the ability to analyze poems. Once it isn’t relevant, then your chances of analyzing the poem correctly—yes, I’m expressing this a bit crudely—but your chances of analyzing the poem correctly don’t change because you did all kinds of experiments in physics and studied physics. It doesn’t, because the information you went through and accumulated isn’t relevant. So learning from experience is relevant only for future cases that are connected to the cases you learned from in your experience. Or, in other words, cases between which there is dependence. But if there is no dependence between them, then the experience you accumulated, or what happened until now, teaches you nothing about what will happen from here on. And that is exactly the relation between absolute probability and conditional probability. The a priori and the a posteriori look exactly the same. In Bayesian inference it’s not like that. The a priori is what you would have thought in advance: what’s the chance of rolling a five? But if after a long time I reached the conclusion that this die always produces odd numbers, then from my experience I already understand that this is a die that lands on odd numbers with high probability. Okay? Now when someone asks me what the probability is that a five will come up, I’ve learned from experience, so I’ll answer one-third, not one-sixth. That’s Bayesian inference, or learning from experience; basically it’s the same idea. Now this is a very, very simple description, but it has extensions, and Bayesian inference is a whole world unto itself. There are so many fallacies and confusions there, but basically it’s the way we learn from experience, and it turns out that human beings are pretty weak at Bayesian inference. I mean, I don’t know, there are all kinds of theses as to why. Evolution didn’t prepare us for it, did prepare us for it, I don’t know, but there are various interesting theses about this matter. But that’s why Daniel Kahneman made a good living out of it. The kinds of fallacies that happen to us when we don’t learn from experience. We don’t decode correctly the things given to us and we don’t draw from them the right conclusions for the next question we’ll encounter. And the Monty Hall problem, all these things are the same. Also, yes, what someone mentioned earlier—or maybe didn’t mention, I thought someone mentioned—the case of Munchausen syndrome. Yes, I talked about the woman in England whose two children died of crib death, and the doctor came—yes, Sir Roy Meadow—came to court and testified that 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, and therefore obviously it didn’t happen by chance. The woman murdered them. They put the woman in prison on the basis of this statistical consideration. Fine? Now there are so many fallacies here that the ink would be exhausted and the paper would be exhausted and we still wouldn’t manage to write all the… But one of those fallacies—and not the central one—is that there may very well be dependence between the two events. It is not correct to multiply the probabilities, because we don’t know why babies die of crib death, but it’s entirely possible that there is something in the family or in the family’s genetics that caused this death. It’s a death whose source we don’t know, but it may have a genetic source, maybe yes, maybe no; but if so, then the probability that a second child will die is no longer one in eight thousand. Remember the example of rolling a three twice? The die falls on three twice. If there is some reason why a three came up the first time, and that same reason is also present in the second roll, then the probability that the second roll will also produce a three is high; it isn’t one-sixth. Same thing here: if there is some cause in this family because of which the child died of crib death, then certainly the probability that the second child will die of crib death is no longer one in eight thousand; it is much, much higher. So first of all, you couldn’t multiply the probabilities. Even if you could multiply them, it changes nothing, but first of all you also can’t multiply the probabilities. And again, the whole question begins and ends with whether there is dependence between the events. Is the second death an event independent of the first death, or is it dependent on the first death? Did the first death teach me something about the second death, or is the first death an irrelevant event from which I can infer nothing about the second death? Yes, by contrast there is, for example, the shell-crater myth, right? You’re under bombardment, and there is this myth that says it’s advisable to hide inside the crater of a previous shell, a shell that already fell. Why? Because the probability that two shells will fall in the same place is even lower. So if one shell already hit that spot, then it’s best to hide in that crater because there’s no chance the second shell will hit the same place. Which is of course not true. The probability that the shell will hit the same place is completely equivalent to the probability that it will hit any other place. There is no difference at all, because there is no dependence between the two events. Between the distribution of the first shell and the distribution of the second shell, there is no dependence. Like rolling a die.

[Speaker E] That was probably Home Front Command guidance too, no? What? That was probably also Home Front Command guidance.

[Rabbi Michael Abraham] I’m giving you a task: find us the relevant diagram. Yes. In any case, Uriel is the one who found the diagram I sent you about standing, sitting, kneeling, kneeling with a kuf. In any case, there really there is independence between the events. And once there is independence between the events, then the fact that a shell fell here once has nothing whatsoever to do with the probability that the second shell will also fall here. The second shell, once again, is distributed uniformly over the area, and therefore it can fall here or anywhere else with the same probability. And the fact that it fell here once changes nothing at all about the probability next time. This is very, very unintuitive, because it’s true that the probability that two shells will fall in the same place is low. That is certainly true. So why, if it already fell there once, is it not worth hiding there? Because the probability for any two concrete places is low. Not specifically for the same place. Meaning, the probability that the first shell will fall in location five and the second shell will fall in location thirteen is also very small, exactly like the probability that two shells will fall in location five. There is no difference at all. Okay? Therefore the fact—what?

[Speaker F] Actually there’s an information gap in your example. Why? Because it could be that the place where the shell fell the first time is a target for that artillery piece, because they think there’s something important there, and therefore it will fire all the remaining shells there too.

[Rabbi Michael Abraham] No, no, it’s a question of resolution. Let’s say we’re talking about, I don’t know, thirty by thirty meters, okay? Meaning, obviously they’re firing at the thirty-by-thirty-meter compound. That’s what they’re firing at; the unit is there. Okay? So they’re firing at them. Now the question is which five-meter square it will land in. Fine? There are nine five-by-five squares, right? In… oh no, what did I say there? Thirty by thirty. Thirty by thirty means thirty-six five-by-five squares. Fine? So now the question is in which square the shell will land. And the assumption is that it is distributed uniformly among those thirty-six squares. Okay? So it changes nothing. Of course if the resolution is larger—if you’re talking about… obviously it’s better not to be in a place that is a distinct target than somewhere else. Or if the artillery’s error really isn’t uniformly distributed but rather has some very specific bias, then if it missed there once probably next time too it will miss to the same place. But if I say under the assumption that the distribution is uniform, the probability of two hits in the same place is still very small, and nevertheless there is no reason to hide in the shell crater. That’s basically the claim. Yes, it’s like many times in basketball, if you know, say they’re shooting three-pointers. Okay, now sometimes there’s a bad night, right? The threes just aren’t going in. Meaning, the percentage of three-pointers made is low. Okay? Now suppose at the beginning—in the first half or the first quarter—the team or player, doesn’t matter, usually has, I don’t know, thirty-five percent from three. Fine? Based on accumulated experience. Now in the first quarter that thirty-five-percent shooter made only fifteen percent of his threes. So people always say, the statistics will even out. Meaning, in the next quarter he’ll probably shoot fifty percent.

[Speaker C] Regression to the mean. What? You’ll get regression to the mean.

[Rabbi Michael Abraham] Exactly. So, regression to the mean. Now, regression to the mean is a bit tricky; the question is how you define it. But in the simple sense, the statistics are not supposed to even out. Meaning, you can’t—you can’t assume that if now it was fifteen, then afterward it will be whatever it needs to be in order to make it fifty-five. Even to end up at thirty-five by the end, the average. No, somehow over the long run it will get to the average, but the probability of missing the next shots is the same probability as it was even without the low percentage earlier, or with the low percentage earlier. Your ability to score is distributed uniformly in the same way, and it doesn’t matter what happened earlier. And still, the law of large numbers says that in the end you’ll reach the average. Now regression to the mean is very tricky. Sometimes people apply it incorrectly, sometimes correctly, but it’s trickier. Okay. In any case, this is, for example, a situation where, yes, like with the shells, there is no dependence between the events. Meaning, the question whether you’ll miss or not miss on the next shot doesn’t depend on whether you missed or didn’t miss on the first shot. It’s exactly like with the shells. But in a place where it does depend—for example, if you’re missing, that may mean that today you’re tired or unfocused or something like that—then in fact it may mean there’s a higher chance you’ll also miss in the next quarter. Not regression to the mean but the opposite. Meaning, you’ll continue shooting worse, because apparently today you’re in some less focused mode or something like that. So that is a place where there is dependence. Where there is no dependence, then no—the distribution is the same distribution regardless of what happened in between. Okay, so that’s enough of a not-so-short introduction about statistical dependence between events. Now I return to double doubt. In double doubt we require that the double doubt be reversible. Meaning, that it be possible to formulate doubt A and then doubt B, and also doubt B and then doubt A—that the order of the doubts should not matter. What does that actually mean? Try now to think in terms of what we just discussed. It basically means that the occurrence of event A, yes, does not depend on event B, and vice versa. Right? The fact that I can first present event A and then event B, or first event B and then event A, basically means that the order of occurrence doesn’t matter, and that won’t change the calculation. If the order of occurrence doesn’t matter, that basically means they are not dependent. The probability that she had relations before marriage or after marriage is one distribution. The probability that it was coerced or consensual is a second distribution. There is no dependence between them. Whether it was coerced or consensual, before marriage or after marriage—there is no dependence between the events. Once there is no dependence between the events, the double doubt is reversible. You can assume it was through coercion or consent, and even if you say it was by consent, maybe it was before marriage and not after. And you can be in doubt whether it was before marriage or after, and even if it was after, maybe it was through coercion. You can formulate both kinds of doubts. Meaning, there is a kind of independence between the doubts. Since that is so, you can basically multiply the probabilities, okay? Because they are not dependent. So in such a case the double doubt is indeed effective and you can multiply the probabilities. But if there is dependence between the events—that is, if the order of occurrence matters—yes, if you tell me that a three came up and you ask me what the probability is that the next one will be a three, that means the order of occurrence matters: if a three came up first, that affects the probability that a three will come up afterward. Therefore, in such a case there is dependence between the events. If there is dependence between the events, you can’t multiply the probabilities by one another. You need to speak about conditional probabilities. In the common case, by the way, if there is dependence between the events, that means the second event has a higher probability if the first event occurred. Higher, specifically. Not always, but in most cases that’s how it is. And therefore the double doubt is weaker. A weaker double doubt. And since it is weaker, they tell me: fine, if the double doubt is reversible then we do not—we do not rule leniently with it as double doubt. Now here one has to be careful. Up to this point I’ve been speaking in slogans, but when we return to the case before us, look: where was the case in which the double doubt is not reversible? I’ll remind you. It was the doubt whether the husband is expert regarding an open entrance or not expert regarding an open entrance, or all the more so perhaps he simply did not identify correctly and in fact the entrance was not open at all. And we have a doubt whether it was through coercion or consent. Okay? Now, we said there is dependence between the events, meaning they are not reversible. Why? Because if you tell me: I have a doubt whether the husband was expert regarding an open entrance or was not expert regarding an open entrance, fine? And even if he was expert regarding an open entrance—yes, so that we know she had relations—there is still the question whether it was through coercion or consent. That is fine. Now I have a doubt whether it was through coercion or by consent, and even if you say it was by consent, perhaps he is not expert regarding an open entrance. If it was by consent, then there is an opening here, so there is no question whether he is expert or not expert; I’m assuming he is expert. Otherwise I couldn’t have asked that question. This order of presenting the doubts is not possible. Only the first order. I can first doubt whether the husband is expert or not expert, and then doubt whether it was through coercion or consent. I cannot first doubt whether it was through coercion or consent, and then doubt whether he is expert or not expert, because the doubt whether it was through coercion or consent assumes that there is an opening here, so there is no room now to doubt, wait a second, maybe there wasn’t any opening at all. Okay? Therefore this is a double doubt that is not reversible; that is what the old Tosafot in Ketubot say. Now notice why this double doubt is not reversible. Just a second. This double doubt is not reversible, but that does not indicate in any way whatsoever dependence between the events. What dependence is there between whether the husband is expert or not expert, and the question whether it happened through coercion or consent? If she had relations, then she had relations either through coercion or consent. There is no dependence between the events at all. Let’s formulate it this way. Suppose I formulate it in a way I’m allowed to formulate it, namely: a doubt whether the husband is expert or not expert, and even if he is expert—meaning there was an opening there—the question is whether it was through coercion or consent. That is the legitimate formulation. The reverse formulation is not legitimate. This is the legitimate formulation. Now I’ll ask you: what is the probability that the woman is truly forbidden to him? Twenty-five percent. It is the product of the probabilities. Why? Because the probability of whether the husband is expert or not expert, fifty-fifty, is in no way connected to the doubt whether it was through coercion or consent, right? The doubts really are independent. The reason I cannot switch their order is because of the content of the doubt, not because of dependence between the events. Simply because of the content of the doubt: if you assume there was an opening, you can’t then doubt whether there is or isn’t an opening; you already assumed there is. So it’s just a problem in that you have to formulate this pair of doubts in a certain order, because the reverse formulation simply cannot be stated that way in terms of the content of the events. Okay? But it’s not really because there is dependence. There is no dependence between the doubts. The question whether there was coercion or—let’s put it this way: does the fact that I know the husband is expert regarding an open entrance change at all the distribution of whether it was through coercion or consent? I’m talking about statistical dependence as I defined it earlier. I know the husband is expert regarding an open entrance; now does the distribution of whether it was through coercion or consent change? Is it no longer fifty-fifty? No, there is no connection at all. I cannot formulate the doubts in the reverse order simply because in terms of the content of the doubts it cannot be formulated in that way. It can only be formulated in the first way. But still, if you calculate the probabilities according to the first way, the result will be the product of the doubts. It is still the product of probabilities. I remind you that this whole series, this entire segment in the series, deals with the product of probabilities, probabilistic multiplication. So this is indeed the product of probabilities, and therefore here the non-reversibility does not stem from dependence between the events. It is true that when there is dependence between the events, then a double doubt will not be reversible. That is true. But it is not true that if the double doubt is not reversible, that means there is dependence between the events. That is not true. A double doubt may fail to be reversible because of reasons of content, not because of a connection between the events, but because you cannot present the content of the doubts in that way. That’s all. If you are in doubt whether it was through coercion or consent, then you have already assumed there is an opening. You can’t now suddenly begin doubting whether there is or isn’t an opening. You must first doubt whether there is or isn’t an opening, and even if there is an opening the question is whether it was through coercion or consent. That is the only way you can present this doubt, but the problem is a problem of language; it is not because there is real dependence between the events. There is no dependence between the events. Therefore the explanation that is apparently very attractive—that you need dependence between the events in order to multiply the probabilities, and then double doubt really expresses a low probability of prohibition—that explanation is not correct for this case. Because in this case the non-reversibility does not express dependence between the events. There is still no dependence between the events, and the double doubt has low probability just like a reversible double doubt, the same thing.

[Speaker A] But in this case we don’t have a double doubt.

[Rabbi Michael Abraham] Why not? We do.

[Speaker A] There’s one doubt. No. If a person is not expert regarding the opening, then it no longer matters whether it was through coercion or consent. He simply doesn’t know, so there is one doubt here. There is no double doubt here.

[Rabbi Michael Abraham] No, the opposite. What you just said is exactly a double doubt. You have a doubt whether he is expert or not expert.

[Speaker A] No, if he—I know he is not expert.

[Rabbi Michael Abraham] No, wait. If you have a doubt—no, I’m talking about when you don’t know. You don’t know whether he is expert or not expert. Even on the assumption that he is expert, and then if he says there was an opening there, apparently there really was an opening there, you are still in doubt whether it was through coercion or consent.

[Speaker A] But again, I assume—my assumption causes this to be not a double doubt but one doubt. Meaning, if he is expert, then there is an opening here, so my doubt is consent or coercion.

[Rabbi Michael Abraham] But if he is not expert, then there is no opening at all.

[Speaker A] So again, then there is no question of coercion or consent. You understand? The question falls away.

[Rabbi Michael Abraham] No. Every double doubt is like that. Every double doubt means this: if option A is correct, then it is permitted. If option B is correct, then it is forbidden, but no—only if such-and-such happens and not if such-and-such happens, then another doubt arises. On the other side, the doubt is irrelevant. It’s always like that.

[Speaker A] So why do we call it a double doubt?

[Rabbi Michael Abraham] Every double doubt is like that—on the contrary. Every double doubt is like that. A doubt whether it happened while under him, a doubt whether not under him; a doubt whether through coercion, a doubt whether by consent. That is reversible, right?

[Speaker A] Right. Good.

[Rabbi Michael Abraham] Now, if it was not under him, do I care whether it was through coercion or consent? What difference does it make?

[Speaker A] Then that also isn’t a double doubt.

[Rabbi Michael Abraham] It is a double doubt. It is fully a double doubt. That is the example the Talmud gives for double doubt.

[Speaker A] The question—

[Rabbi Michael Abraham] Whether it was under him or not under him, and even if it was under him, the question is whether through coercion or consent.

[Speaker A] Right, but one side always drops out.

[Rabbi Michael Abraham] Which side? No. You can doubt whether it was through coercion or consent, and even if you say by consent, it may have been under him and it may not have been under him. You can formulate it. It is completely reversible. Completely reversible, and the events are not dependent, and it is a double doubt. Fine. But the doubt about whether he is expert or not expert regarding an open entrance is a double doubt that is not reversible, and yet the doubts are still independent. The probability is the product of the probabilities. Therefore non-reversibility is some indication of dependence, but it is not true that if there is non-reversibility then clearly there is dependence. If there is dependence, there is non-reversibility. But not that if there is non-reversibility, then there is dependence. And therefore what I actually want to claim is a weaker claim. This explanation is not as good as I thought at first. My claim is this: we are speaking about negative doubts. We do not know how to perform the calculation. We want to establish some rules detached from probability calculations, because we do not know how to do the probability calculation. We want to establish some rules that will maximize the chance of permission, right? Or increase the chance of permission beyond one doubt, beyond a simple doubt. So what am I saying? I’m saying this: look, if the double doubt is not reversible, then maybe that is because there is dependence between the events, and then indeed there is no justification for relying leniently on double doubt. It is true that there are cases in which the double doubt is not reversible and this does not express dependence—sorry, dependence. But since very often non-reversibility does express dependence, I establish a sweeping rule. So I basically say: look, once the double doubt is not reversible, we are not lenient on the basis of double doubt. Because—call it “we make no distinctions,” yes?—because if the non-reversibility expresses dependence, then it is really true that I cannot rely on it. It does not always express dependence, but we are not doing statistical calculations; we want to establish uniform legal rules. So what I say is: if it is not reversible, I already suspect there is dependence here; we do not rely here on double doubt for leniency. Even though in many cases the non-reversibility is non-reversibility because of the content of the doubts, not because of statistical dependence between the doubts. Okay? So it is still an explanation, but not such a successful one as one might have thought at first glance. And by the way, that may be one reason there are quite a few later authorities who do not accept the requirement of reversibility. They argue that even if the double doubt is not reversible, it is still a double doubt. What difference does it make whether it is reversible or not? And on the conceptual level they are right, if the non-reversibility is not the result of dependence, sorry. It is not the result of dependence, as in the example we saw earlier. But if I am right about this, then those later authorities too would admit that in cases where the non-reversibility truly stems from dependence… in those cases they too would agree that this is not double doubt, because they do not require reversibility in principle. Okay? For example, regarding the knife and the neck joint, if you remember the example from last time, regarding the knife and the neck joint: with the knife there is a doubt whether this happened before the majority of the two simanim or after the majority of the two simanim, and the second doubt is whether it happened at all during the slaughter or when it had already reached, after the simanim, the neck joint. Fine? Now there it is not reversible, because if it happened before the majority of the two simanim, then there is no room to doubt whether it happened at the neck joint. It did not happen at the neck joint; it happened before the majority of the two simanim. Okay? Now there, that really—there really there is dependence. It is not non-reversible because there is dependence between the events. If I know that it happened during the slaughter, then clearly it did not happen at the neck joint. Meaning, if it happened before the majority of the two simanim, then clearly it did not happen at the neck joint. Here there really is dependence in terms of content; it is not that I can’t formulate the doubt, it is not a technical problem. It is a problem of genuine statistical dependence. Therefore in that case, the fact that the doubt is not reversible really does mean that one may not rely on it, even according to those who disagree with the Shakh and all those who require reversibility. That is what I want to claim. Except that this particular example, as I already said, also satisfies the criterion that it is one type of doubt. And therefore the fact that it is not reversible is not so interesting, because there too it is one type of doubt. So whatever happens—even if you disagree about reversibility, here you would agree, but not because of my reasoning, namely that when there is non-reversibility due to dependence… rather no, you would agree because it is one type of doubt. Therefore you are not lenient. The question—and I don’t currently have an example of this, I couldn’t think of an example—but I see no reason there shouldn’t be one: is there a case in which there is a double doubt that is not reversible, but there is no issue of one type of doubt here, it is only non-reversible, and the non-reversibility is because of statistical dependence and not because of the substantive content of the doubts? Such an example would be an excellent example for conveying the point. I couldn’t think of such an example when I was thinking about this matter, but I don’t see any reason there shouldn’t be one. I assume that if I think more, I’ll find one. Meaning, I’m not…

[Speaker A] Wait, Rabbi Ovadia Yosef in his responsum writes that there is no need for reversibility. There he has a responsum regarding double doubt being reversible concerning Yoreh De’ah, where it is ruled that… there is a double doubt either in the slaughter or in the neck joint. Does Rabbi Ovadia rule that it is permitted to eat it?

[Rabbi Michael Abraham] I have no idea. But I said…

[Speaker A] But what is the practical Jewish law ruling?

[Rabbi Michael Abraham] I have no idea. I’m saying that there too there is one type of doubt, so in any event one is not lenient there, regardless of reversibility. Therefore that example would not be a practical difference, because even one who disagrees with the requirement of reversibility, there he would not be lenient on the basis of double doubt because it is one type of doubt.

[Speaker A] And regarding expertise—if he found an open entrance, if he is not expert…

[Rabbi Michael Abraham] No. In the Tosafot we saw, because it is not reversible, it is ruled there that it is not—that it is not double doubt. We are stringent. The only question is what the explanation is. The old Tosafot explain: because it is not reversible. Those who disagree with the requirement of reversibility will offer other explanations there; it doesn’t matter right now.

[Speaker A] But do they find explanations toward permission or toward prohibition?

[Rabbi Michael Abraham] No, in the Talmud it says that it is for prohibition.

[Speaker A] Right, the Talmud says prohibition. Now if we say that here there is a reversible double doubt in the first case, yes? That in the case where there is an opening, and whether it was under him or not under him, then we get—sorry—if he is not expert… if he is not expert then it is irrelevant; if he is not expert then there is no opening. So then it must be toward permission.

[Rabbi Michael Abraham] No, that is what Tosafot asks: why shouldn’t it be toward permission, since it is a double doubt? But the Talmud says toward prohibition. You are raising Tosafot’s question.

[Speaker A] That’s what I’m asking: how does Tosafot’s question conclude? What does he answer?

[Rabbi Michael Abraham] Tosafot itself answers that it is not reversible; therefore it is toward prohibition. But one who disagrees with the requirement of reversibility will explain it differently, doesn’t matter, but he will still think it is toward prohibition because the Talmud says it is toward prohibition. Fine. In any case, that is regarding the explanation of… wait. Now just a few remarks about conditional probability. What does conditional probability actually mean? So look… I’ll share the—just so we can see it from a slightly different angle. Look, suppose we have here six outcomes of rolling a die, okay? Now when I ask the question what is the probability of getting a five? One-sixth. But if I am given that the outcome is odd, what does that actually mean? Notice that my space is the space inside this line; it is no longer the whole ellipse, right? Rather only this space. See? Then the result is one-third, because there are only three cases, only three possibilities. Meaning that conditional probability, what it basically does, is shrink the event space. If I have more information, more information means there are fewer events consistent with that information, right? We once talked about information theory, and I said that if I describe Ben-Gurion, for example, as the first prime minister of the State of Israel, okay? Then “the first prime minister of the State of Israel” is a statement rich in information. I know he was the head of something, and that something was a government, and he was the first prime minister, and the first government of the State of Israel. Every piece of information I remove here—for example, the first prime minister of a state, not necessarily Israel—you understand that there are more such people, right? More such people. And if I remove another piece of information—he was the prime minister, the prime minister of a state, not necessarily the first—there are even more such people, right? I say he was a head, not necessarily of a government—there are even more such people. Okay? Meaning, the less information I have, the more people fit that information. If I have more information about the thing, that means there are fewer objects or fewer people described by that information. That is information theory in a nutshell. The more you have, the more intersections you create in its Venn diagram.

[Speaker C] What? You create intersections in its Venn diagram, as though you create more cuts.

[Rabbi Michael Abraham] Yes, exactly. What I drew here is basically a kind of Venn diagram. So basically what I’m saying is that when I add information for you—and I’m talking about relevant information—that means I am essentially cutting down the number of events in the event space, in the sample space, right? Instead of six events, I now have only three. Meaning this is also a way to look at conditional probability. Conditional probability means: what is the probability of B given A? I say: if A is known to me, I have certain information, then that means that now when I ask what the probability of B is, I’m asking what the probability of B is if I am in this half of the ellipse and not in the whole ellipse. And therefore the calculation will change. In this case the probability will increase. I said that usually it increases. When you shrink the space and ask what the probability is that one event will occur, then within a more limited space the probability is greater. There are other situations, but this is the typical one. Okay? So this is basically another way to look at conditional probability: it is essentially a reduction of the event space. Now I’ll give you another example, the presumption that what is in a person’s possession is his. The Talmud says, yes, there is the presumption of possession—sorry, the Talmud says that there is established possession. What does that mean? “One who seeks to extract from another bears the burden of proof.” Right? Meaning, if I sue you, you are the one in possession, and the burden of proof is on me. Now there are later authorities who want to explain this on the basis of the presumption that what is in a person’s possession is his. If you are holding the thing, there is a presumption that it is yours, and therefore if I claim the thing from you, I am at a disadvantage; I am the one who needs to bring evidence. They are basically claiming that possession is based on probabilistic reasoning. Yes? That basically most likely the object is yours if it is in your possession. But that is a mistake. Why is that a mistake? There is a presumption that what is in a person’s possession is his, but that

[Speaker G] has nothing to do with the majority of humanity. What? The majority of humanity, not with people who are litigating.

[Rabbi Michael Abraham] When we talk about conditional probability, what does that mean? When we look at all the objects in the universe, and we ask ourselves how many of them are in the possession of their owners—I already gave this example—how many of them are in the possession of their owners? Say the answer is ninety percent. Okay? Ten percent. So there is a presumption based on the majority that what is in a person’s possession is his. Most objects are with their owners; most of what a person has is owned by him. Okay? Ninety percent, let’s say. Now I ask: does that mean that when I sue you and you are holding something, it is most likely yours? My answer is no. Not necessarily. Why? Because when you are now being accused—really, when I am talking about objects that are under dispute. A legal dispute is a subset of all objects. In that subset, who says that the majority are with their owners? What you are implicitly assuming is that most plaintiffs are liars and only a minority of defendants are liars. There is no reason to assume that. Everyone has a presumption of credibility. And because of that, the presumption of credibility teaches us that the sample of objects that are under dispute is not a representative sample. Its distribution is not like the distribution among all objects. So let me show you this now in exactly the same way we saw the die earlier. Look, here is the same picture. Let’s say these are all the objects in the world. Okay? There are, I don’t know, eighteen objects here, I think I drew. These are all the objects in the world. Four of them are not in the possession of their owners. Sorry—four objects are objects that are under legal dispute. Those are the shaded ones, right? Object one, object three, five, and four. Okay? Those are the objects under legal dispute. The white objects are not; there is no issue there. Now I say: this is the line dividing between objects that are with their owners and objects that are not with their owners. Okay? Now here, notice, on this side of the line there are ten, right? On this side there are eight. Okay? So if you ask where most of the objects are with their owners—ten out of eighteen. That is more, but it is just for the sake of the example, right? But now when I focus only on the group of shaded objects, what happens? Among the shaded objects, only one is with its owner, and three are דווקא with the other person. Meaning, this set of four is not a representative sample of all eighteen. The distribution within the set of four does not represent the distribution within all eighteen. And that is exactly the meaning of conditional probability. Given that this object is under legal dispute, what is the chance that the person holding it is its owner? The probability comes out completely different from the absolute question: the object is in my possession, what is the chance that it is mine? That is a completely different question. The second question, sorry: if the object is in my possession, what is the chance that it is mine? Look at all the objects—ten out of eighteen are with their owners. Okay? Now I say: given that the object is under legal dispute, what is the chance that it is with its owner? Here I need to look—the relevant space is only the four shaded ones. I should really have drawn another sort of circle around the four shaded ones, isolating them. You can see that they are already divided one against three. And דווקא the side that is the overall majority is the minority among the shaded ones. So if I focus on the shaded ones, then there is actually a majority of things that are not with their owners. Just an example, of course, but this is a possible situation. There is no reason to assume that objects under legal dispute are a representative sample of all the objects in the world. On the contrary, there is a good reason to assume they are not. Because both the plaintiff and the defendant have a presumption of credibility, there is no reason to assume that one lies more than the other. And because of that, the distribution is probably really two against two, say. I should have drawn the line somehow like that—two against two. And therefore I cannot take the money away; therefore I cannot express the principle of possession based on the fact that most objects are with their owners, because that majority deals with all objects and not with the narrower group of objects that are under legal dispute. So you see that once again I have narrowed the event space, which really means that the question here is one of conditional probability and not absolute probability. Given that the object is under legal dispute, I have information about it. Remember Monty Hall? I have some information about this object, and that information is relevant. And since I have information, I cannot use the general distribution. I need to do the calculation given the information. So this is another example of the confusion you end up with if you mix up ordinary probability and conditional probability. Maybe one final remark: this also applies to scientific generalizations. I have also spoken about this already, but I am just tying things together now. In scientific generalizations, we too are basically learning from a sample… to a general case. Say I saw several massive objects falling toward the earth, and I infer the conclusion that all massive bodies fall toward the earth. How can I infer from the narrow group to the broader group? Only if I assume that the narrow group is a representative sample. What does representative sample mean? It means that the distribution in the small group looks like the distribution in the large group. There is no statistical bias created by narrowing the group. Yes, say, if I go back for a moment to the presumption that what is in a person’s possession is his—yes, if for example there had been nine here and nine there, say move this white one over here, then it would be nine and nine, and the line would pass here. Meaning, these two would be on the right side and these two would be on the left side. Then the black ones would be a representative sample, right? Because the black ones would be distributed half and half, two against two, and the overall set would be distributed nine against nine. So the distribution within the group of four would look like the distribution within the group of eighteen. In that case, the group of four is a representative sample. It is a sample that behaves like the whole group; it represents the whole group. Its distribution is like the general distribution. A scientific generalization always assumes that the cases we encountered are a representative sample of all cases, because otherwise you cannot make a scientific generalization. And in places where the sample is not representative, making a generalization is a fallacy—which is exactly what we did with the presumption that what is in a person’s possession is his. We made a generalization from a non-representative sample, and therefore it is a fallacy. Okay, let’s stop here. Next time I think we will already finish; maybe we will need one more time, but I have said that before. Still, it seems to me that here we are already closer to the correct assessment. I still basically have two more points. I have two more Talmudic cases in which I will show you how narrowing a majority by means of conditional probability changes the situation, and this is in the Talmud, in two Talmudic cases. And after that I will speak about a responsum of the Havot Ya’ir, also an interesting statistical problem, and with that we will apparently finish the series. That is it. Comments or questions?

[Speaker D] Rabbi, you began today’s lesson by saying that we are talking about negative doubts. We do not really know the statistical likelihood of each side, but if we knew it was 50%, since we are talking about a Torah-level prohibition we would rule stringently. But since this is a negative doubt, we hope to arrive at some kind of majority.

[Rabbi Michael Abraham] No, no. In a negative doubt we are also stringent. In a negative double doubt we are lenient.

[Speaker D] Yes, yes, with a double doubt, exactly. Now I am asking: suppose there were some divine revelation that said that what is at stake—meaning, if we violate the prohibition here and it really is prohibited, it is poison, and there would be some general collapse of all the spheres and the world would return to chaos. Would we still say negative double doubt? What? It is poison and not a prohibition.

[Rabbi Michael Abraham] It is poison and not a prohibition. Yes. If you have a double doubt regarding poison, are you lenient there too?

[Speaker D] No, exactly not. Obviously. So then how does the Rabbi explain it? What is the problem? If the prohibition really is significant, we treat a prohibition as no less than poison.

[Rabbi Michael Abraham] It is significant, but the Torah decided to be lenient

[Speaker D] toward us, it does not

[Rabbi Michael Abraham] require us to be stringent.

[Speaker D] But how did it decide that? I am asking—there is a contradiction; it cannot contradict itself. If the Torah tells us there is a prohibition, the Torah tells us that we damage the eternal within the hod, and on the other hand it tells us that in a positive double doubt we follow the majority, and in a double doubt even in a negative doubt we follow the majority. So how do those go together?

[Rabbi Michael Abraham] First of all, the question is how severe the damage is, how reversible it is, the question of our trust, the question of whether the Holy One, blessed be He, supports that decision and prevents the metaphysical damage.

[Speaker D] The Rabbi kept telling us that even the Holy One, blessed be He, cannot contradict logic. If logic determines that this Torah-level prohibition will cause damage… What does logic have to do with it?

[Rabbi Michael Abraham] What does this have to do with logic?

[Speaker D] Why not? If it is reality… what do you mean? If the prohibition is not just a prohibition in the sense of angering the

[Rabbi Michael Abraham] Creator, but a real injury to the eternal within the hod, then the Holy One, blessed be He, prevents it from being cut off. What contradicts logic here? I do not understand.

[Speaker D] So there is not really damage? The Holy One, blessed be He, does not think there is damage?

[Rabbi Michael Abraham] That is one possibility. A second possibility is that this damage, with all due respect, but if I am stringent in these cases of doubt then I too suffer great damage, and the Holy One, blessed be He, allows me to avoid that damage even at the

[Speaker D] expense of the damage to the eternal within the hod.

[Rabbi Michael Abraham] No, but it could be that if they do not permit me—if they do not permit me to eat an egg that fell into a hundred prohibited eggs, they do not permit me to eat it—that means I now throw a hundred eggs into the trash even though all of them are kosher except one. Sorry, a prohibited egg that fell into a hundred permitted ones. If they do not permit me to eat it, that means I am now throwing a hundred eggs in the trash even though all are kosher except one. So that is unreasonable, and so the Torah wants to be lenient with us; it says okay, you are allowed to be lenient.

[Speaker D] But how can it be lenient with us about something that has significance in reality?

[Rabbi Michael Abraham] Such an egg also has significance in reality. What do you mean—what is the problem?

[Speaker D] So then how can it be permitted?

[Rabbi Michael Abraham] Because the financial loss of a hundred eggs also has significance in reality. Right, but right now we are speaking not about businesses selling leavened food, and private individuals are not permitted to sell actual leavened food.

[Speaker D] I do not know, it is permitted for private individuals too, and private individuals sell it.

[Rabbi Michael Abraham] Businesses have substantial loss.

[Speaker D] Yes, but if private individuals sell and seriously intend to sell, then certainly it takes effect, and then there is no problem at all.

[Rabbi Michael Abraham] No, but then you have a concern that the sale is not serious, and if you have such a concern that undermines it.

[Speaker D] No, but if the person himself intends it—no, fine, but the fact is that it depends on the sale; they do not sell seriously, not really. Shmuel, here too it is substantial loss,

[Rabbi Michael Abraham] Shmuel, here too it is substantial loss.

[Speaker D] I am asking the Rabbi, I am asking: how can it be that we categorically permit all severe and difficult and terrible Torah prohibitions without concern that we are eating a sacrifice because of some negative doubt, on the basis of a double doubt?

[Rabbi Michael Abraham] Not the most severe and terrible prohibitions. No one will permit you to murder on that basis, and no one will permit you to commit adultery on that basis.

[Speaker D] We saw the permission regarding agunot, and we saw the kind of hysteria there is there; it is very far from a double doubt. No double doubt would permit that. What I think the Rabbi understands I mean is that all this hints that in the end it is not the statistics that determine things, but the significance of the phenomenon. And since Torah prohibitions between man and God are not prohibitions that stand in the eternal within the אות, but are really connected to their effect on us, then we can exercise judgment. Regarding murder, the Rabbi is right that we would not do that; we would not follow a double doubt.

[Rabbi Michael Abraham] I will even answer you with what I am expected to answer you: everything is foreseen, yet freedom of choice is given.

[Speaker D] Maybe I disagree with that choice.

[Speaker H] Rabbi, isn’t an egg a complete entity? What? Isn’t an egg a complete entity? I can’t hear. An egg—an egg isn’t a complete entity?

[Rabbi Michael Abraham] It does not matter, it does not matter. Okay, fine, let’s leave that because of the example. On the principled level, yes.

[Speaker A] Ah, Rabbi, in civil law, what…

[Rabbi Michael Abraham] A complete entity as a reason not to be nullified is also rabbinic. What are you saying? Yes.

[Speaker A] Ah, regarding what we just discussed, about “the burden of proof rests on the one who seeks to extract from another,” in Torah law, yes, we always have this rule; it is supposedly fundamental, yes. But in the legal system… whenever people come to a religious court, then the burden of proof rests on the one who seeks to extract from another. Now in Hebrew law in the state there is also the principle that the thing speaks for itself; that is what he just said, what you just showed in the picture, that if I have proof that the thing is, as it were, not in the owner’s possession, then it flips. So it becomes that the burden of proof is on the defendant and not on the plaintiff.

[Rabbi Michael Abraham] If you bring evidence, then you extract from the one in possession. Okay, so what?

[Speaker A] The question is, I do not remember in Choshen Mishpat whether I saw this—does Torah law also have this rule?

[Rabbi Michael Abraham] Of course. “The burden of proof rests on the one who seeks to extract from another” means that you have to bring the evidence, but if you bring evidence…

[Speaker A] No, if I place in… “the claimant shall approach,” how is it written in the verse in the Torah? “Whoever is the claimant shall approach them.” Yes, “whoever is the claimant shall approach them,” meaning always

[Rabbi Michael Abraham] Whoever is the claimant has to approach.

[Speaker A] Right, but when he brings the evidence he

[Rabbi Michael Abraham] will receive the money even though he is not the one in possession. “The burden of proof rests on the one who seeks to extract from another” means that the plaintiff has to bring the evidence, but it also means that if he brings evidence then he wins even though he is not the one in possession. Evidence can extract from the one in possession; you just need to bring evidence.

[Speaker A] No, “the thing speaks for itself” is also

[Rabbi Michael Abraham] evidence? I do not know what “the thing speaks for itself” means. A presumption? Yes, a presumption is evidence. “A person does not repay before the due date” in tractate Bava Batra. A person does not repay before the due date—that is evidence; it reverses the burden of proof. The one in possession will have to bring evidence if he claims that he repaid before the due date. Even though today people do in fact repay before the due date

[Speaker A] a lot. We are talking right now when the assumption is that people do not repay before the due date. Is there a סעיף for this in Choshen Mishpat? Of course. What, which סעיף?

[Rabbi Michael Abraham] Look it up. In Bava Batra 5a, in Ein Mishpat. More power to you. Okay, Sabbath peace, goodbye. Sabbath peace, thank you very much.

Doubt and Probability—in Halakha, Thought, and in General—Lesson 49—Rabbi Michael Abraham

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