Induction – Continued: On accidental significance (column 730)

In the SD

In the previous column, I dealt with generalizations in halacha and in general, I discussed the problematic nature of generalization procedures and the ways that are taken to avoid them. We saw the meanings of intuition and a priori explanation on the subject of generalizations. The discussion was conducted through the issues that deal with the assumption of three times. One of the examples given in the Gemara for the assumption of three times is a situation in which a woman has married three times and each time her husband died immediately after the marriage. In such a situation, we prohibit her from marrying again, and according to the opinion of Rav Ashi (Yivamot 67b), who also ruled in halacha, this stems from the fear that her luck will cause her husband to die, and therefore there is a fear that her next husband will also die. I explained there that this is a fear and not a positive conclusion, and the basis for the fear is statistically significant findings. The recurrence of a case three times is significant enough to create concern.

One of the implications of our a priori assumptions in the generalization process is that where we do not have a reasonable explanation that interweaves the three cases, we do not generalize. I further explained that Rav Ashi’s concern that luck was the cause is not a shot in the dark. He is not talking about luck in the sense of randomness but in the sense of the influence of the stars. Following the words of Mekor Chaim in Hal’ Pesach quoted there, I showed that when there is no explanation, we do not generalize. For example, if we find that the successful schools all have names that begin with a letter between א – ח, we would not conclude from this that it is advisable to change the names of the schools in order to improve their achievements. Why? Because a priori it is unlikely in our view that the name of the school is relevant to its achievements. I also argued that in our scientific view today there are no influences of the stars, and therefore if I saw a woman whose three husbands died immediately after marriage (in a way that it is clear that she did not murder them and that they did not die as a result of contact with her), I would not fear that luck was the cause and I would not forbid the fourth from marrying her. I would prefer to say that this was a pure coincidence (meaning that the luck that caused this is ‘luck’ in the modern sense, and not the influence of stars).

This of course raises the question of whether there is no mental fixation here. Ostensibly, there is clear data that shows that luck does have an effect (as you may recall, the halakhic assumption is that three repetitions are not a coincidence), and yet I insist on assuming for granted the scientific assumption that the stars do not have such an effect on our lives, and I fortify myself with this assumption even against clear statistical data. Ostensibly, I refuse to make a generalization even where it is called for. I argued there that such patterns are coincidental and therefore, despite the apparent significance of the data, it is not correct to see them as a good basis for making a generalization, since there is no reasonable explanation that interweaves the coincidences. A priori reasoning rejects statistical significance and leads us to the conclusion that it is still a coincidence.

In this column, I will try to explain the logic behind this approach, and why it is not just a matter of conservatism and fixed thinking. I will offer two perspectives on the phenomenon (which will later appear to be related): the law of small numbers and broadening the perspective.

Explanation A: Law of Small Numbers

There is a statistical phenomenon that is important to consider, which is somehow related to the law of small numbers, even when they are not that small, and from which it becomes clear that even a significant phenomenon can be coincidental.

Consider the example we saw in the previous column, where the highest-achieving schools are small. We saw that it is dangerous to conclude from this that a small school is beneficial for improving achievement. Although I attributed this to the fact that there was a statistical error there, since they did not look at the worst-achieving schools (which were also small). We saw that the two tails of the Gaussian would contain the cases where the numbers are small (small schools or small towns with respect to kidney disease).

But what if we really saw that the ten successful schools are small schools, and the worst are not small. Is it then correct to conclude that a small school is beneficial for improving achievement? This makes sense, but it is still important to understand that there is no situation in which I cannot find some rule that interweaves all the successful schools. For example, that each of them is in an odd place in the list of schools in the country. Or the first letter of their name is between A and H. Or that they all have a fence made of a material other than wood, a principal who is over 1.80 m tall, and so on. There will be no list of schools, or a list of towns with low morbidity, that I cannot explain ad hoc in this way on the basis of some arbitrary parameter. There will always be such a parameter.

In the previous column (in the section “The Difficulty of This Conclusion”) I mentioned column 482 , in which I dealt in general with this phenomenon, and here I will elaborate a little more. Think of the series of numbers …3,5,7. What is the next number? Ostensibly 9, since it seems that this is the series of odd numbers. But I could equally suggest 11, and explain that this is the series of prime numbers (those that are only divisible by themselves and 1. 9 is not prime since it is divisible by 3). So who is right? No one is right here, since both explanations are possible. In the above column, I showed that this is true for any series of numbers you choose, since for any such series, countless rules can be found that explain it, and each such rule will dictate a different additional number. It’s just like I commented about the schools with the highest achievements. You can list them by saying that they are small, or that they have a wooden fence, or that the principal is over 1.80 m tall, and so countless explanation options are all appropriate. I suppose we’ll still choose the explanation that they are all small, since we have a good story behind it. It seems to us that this parameter, unlike all the others, is relevant.

You see, the a priori explanation dictates whether and how we will generalize. In column 715 , we saw this from a different angle, and I explained that without initial intuitions, no scientific generalization is possible. The conclusion is that Kahneman is right in saying that the existence of a good story can lead us to wrong conclusions, but at the same time, such a story is very important when we want to choose between several generalization options. As we saw in the previous column, our a priori perception of the relevance of the explanatory parameter to the phenomena being explained is very important. Without this a priori elimination, we will always have countless explanations for each finding. The mistake that Kahneman points out is that sometimes we choose a good story when there is another story that is no less good, or that sometimes this story biases our statistical analysis (as in the examples of school achievement and morbidity). This, of course, must not happen. But given several generalization options, the story that seems to us a priori to be the best will always dictate the direction of the generalization.

And what if we don’t have a good story? That is, if it is clear to us a priori that the examples we have seen have no connecting line and no generalization that explains them? In such a case, we will sometimes really give up on the generalization even though there is an appearance of significance. Therefore, in the case of a woman whose three husbands died or in the case of an explanation for the achievements of schools according to the first letter of their names, I prefer the interpretation that it is a coincidence and not to attribute it to luck (the influence of the stars) or other strange factors that I have no indication of their existence. This is not a mental fixation but a rational elimination that is indeed based on our a priori assumptions. We must be open to the fact that they are incorrect, but as long as this has not been clearly proven, I would not disparage them.

Explanation B: Broadening the perspective

Let’s look at this from a different angle. The fact is that different people die in different ways and for different and strange reasons (which are sometimes unknown to us and sometimes are). In the review here you will see that some of the poskim speak of the death of the husband at any stage after the marriage (i.e. even if they died of old age after 50 years), but this really seems completely absurd. So for the sake of simplicity we will talk about death in the first year of marriage. This can of course happen completely by chance.

So, let’s assume now for the sake of discussion that we found statistically that in one in every hundred married couples, the man dies in the year after the marriage. In every such case where the man dies, a wife is left behind. In other words, we have about 1% of women in the population whose husbands died in the first year. Now one of them comes along and wants to remarry. Does it make sense to forbid her from doing so? Of course not. The death of one husband certainly does not constitute an indication of concern that the woman’s luck caused the husband’s death, and it makes no sense to forbid another man from marrying her. Even without any special luck on her part, there are natural deaths of men, and it was her misfortune (in the random sense) that such a man fell into her lot. Let’s assume that we have a million married women in the country, then of those, there are about ten thousand women whose husbands died in the first year of marriage. So, when we see a woman whose husband dies, is it reasonable to fear that her luck caused his death and to forbid her from remarrying? Of course not. This is a statistic of male deaths regardless of the woman’s luck. Note that although this is only a 1% chance that such a husband will die, it is clear that the chance that the woman’s luck caused the death is zero. Even without any influence of luck, there are ten thousand such women, and this woman is one of them. This is luck in the modern sense (randomness) but there is no reason to assume that it was an influence of the stars.

Now let’s ask what about a woman whose two husbands died? If her luck was a factor, it means that there is a common cause for the deaths of both her husbands (the woman’s luck). But note that even if there is no influence of luck and we are talking about statistics of men dying in the year after marriage, there will still be a hundred women in the population whose two husbands will die in the first year of marriage. Therefore, if we saw a woman whose two husbands died, there is no reason to think that her luck caused this. She is one of the hundred women who are expected to be in such a situation (out of the million women in the country), and therefore when we meet someone like that, it should not arouse in us any concern or suspicion that this is someone with fatal luck. This is a necessary statistical result, without any connection to the stars or luck.

If we were to see that in some place the number of women who kill their husbands is much higher than the accepted statistics in the world, that’s a different story. And there too, we would have to do regressions to rule out the possibility that this is a coincidence (a small place can produce such results) or that there is another reason that caused this. Alternatively, if a woman comes before us whose ten husbands die in the first year, and when we look around, it seems that there is no woman whose eight, or seven, or six, or five, husbands die, then there is reason to be suspicious (although even there I would prefer to suspect that she killed them rather than that her luck caused it). These are clear statistical indications that can lead us to make a generalization. But in a normal distribution of deaths, it is clear that there will be quite a few women in the population who will have two husbands die, and therefore when one of them comes before us, there is no reason to fear anything. This says nothing about the fate of such a woman’s next husband. The opinion of a rabbi who fears after the death of two husbands of the same woman seems clearly unfounded.

What if three of the woman’s husbands were dead? It seems that here we are already at the statistical limit, since if we are talking about a random distribution of male deaths, we only have a 1 in a million chance that three husbands will die for the same woman during the first year after marriage. In other words, this could happen to only one woman out of a million in the country, but only one. This already establishes some basis for suspicion, and there was room to begin to worry (although not to decide positively, of course). In such a situation, even without any evidence in the three cases, even the Rashbag already warns the fourth woman not to marry this dangerous guy. You can now understand what the basis of the disagreement is between the Rabbi (who creates a presumption after two cases) and the Rashbag (who believes that it is created after three). The question is when does reasonable concern begin, and it is clear that there is no sharp statistical line here.

But I think that with the numbers I gave here, even if three husbands had died, I wouldn’t be worried. At least if there are indeed ten thousand in the population with one dead husband and a hundred with two dead husbands, then when I found one with three dead husbands, there is no reason to assume that this is not the result of the distribution of deaths expected by chance. Here too, I wouldn’t be worried about anything. If there were no women with dead husbands in the first year at all and only one with three dead husbands, all in the first year, that should already arouse suspicion in us.

This is the phenomenon of broadening the perspective. When we broaden our perspective beyond the case before us and examine the distribution in the entire population, we discover that the case that at first glance seems to us to be obvious is nothing more than a statistically expected case. This is the explanation for my conclusion from the end of the previous column that where there is no reason to suspect that there is a common explanation behind all the recurrences, it is more correct to assume that these are isolated cases. This is not stubbornness but rational thinking.

Example from life: Shimon Cooper and his three wives

My good friend, retired judge Menachem Finkelstein, sat at the head of the panel that dealt with a similar case. There was even a book written about it , Catching a Murderer , by Omri Asenheim, who also did a television investigation into the affair (and through him even had some part in the conviction). A man named Shimon Cooper was convicted in 2016 of murdering his first wife (this murder happened in 1994) and of murdering his third wife (which happened in 2009). Initially, the case about the third was closed due to lack of evidence, but then additional evidence was discovered and he was convicted. Part of the conviction for murdering his first wife was based on the fact that he was convicted of murdering the third (when both of them were perfectly healthy and died suddenly). Really strong, except here it happened after one time. Incidentally, Cooper was later also investigated on suspicion of murdering another woman, a Holocaust survivor whom he was suspected of wanting to rape.

What would have happened if his third wife had died? I don’t know if these two murders were enough to convict him of the third. Ostensibly, according to Rabbi, there is a place for this, since Rabbi bases a presumption on recurrence twice. But it is quite clear that this is not true. Rabbi is talking about suspicion and apprehension, not about a positive conclusion. In other words, Rabbi would warn a woman who wanted to marry a given cooper and would forbid her from doing so because of the presumption. But a conviction for murder requires a positive conclusion. In the conversations I had with Finkelstein at that time, we dealt quite a bit with such statistical phenomena. I will now give some of the examples I gave him at the time.

A. The Black Swan

Nassim Taleb, in his book The Black Swan , made a similar point. Think of an exceptionally successful investor like Warren Buffett. I think you’ll agree that he has fantastic skills and understanding in the field of financial investments and evaluating the success of various companies. Surprisingly, Taleb challenges this conclusion, and his argument is as follows.

Consider a market that consists entirely of a collection of investors who all lack any understanding or ability in the field, and each of them decides completely randomly what to invest their money in. For the sake of argument, let’s assume that out of ten such investors, one will consistently make a profit for an entire month. One out of a hundred such investors will consistently make a profit for a year. One in a thousand will consistently make a profit for ten years. One in ten thousand for thirty years, and one in a million will consistently make a profit for a hundred years. Let’s assume, for the sake of argument, that Warren Buffett has been making a profit consistently for about fifty years. Does that mean he has extraordinary talent and understanding? Not necessarily. He could just be a random investor, and there are millions of investors in the world. Statistically, one of them should make a profit for a hundred consecutive years even without any talent. So, it’s possible that Warren Buffett is completely talentless, and his profits come from simply being that one. Taleb’s claim is that Buffett’s luck may have played a role (but not in the sense of the influence of the stars, but in the sense of randomness).

This is of course again the phenomenon of broadening the perspective. Here too, when we broaden our perspective beyond the case before us and examine the distribution in the entire population, we discover that the case that at first glance seems to us to be obvious, may be completely coincidental, in a way that results from a simple statistical calculation. Of course, if we were to broaden our perspective and discover that Buffett is particularly prominent, meaning that there are no investors who have succeeded for forty, thirty, or twenty years, or if he seems to continue to profit for more and more years while all the other players around him drop out of the game, then it may be correct to conclude that his success indicates extraordinary abilities. But in order to reach this conclusion, we must broaden our perspective. With a narrow view that focuses only on the case in question, it is impossible to draw any conclusion. Although there seems to be impressive obviousness here, this could be a case of the past.

By the way, this conclusion can also be challenged. It is possible that even if we find the pyramid of success that I described, what causes it is the distribution of investors’ talents (and not a distribution of random successes regardless of talent). Therefore, statistics of investor success do not necessarily mean what Taleb says. But the possibility that he raises really exists, and that is what is important for our purpose. This requires us to think carefully before we draw conclusions from a chain of events that seems obvious on the surface. Similarly, with regard to the power of three times, there is a possibility that despite the apparently obvious recurrence, it is actually a coincidence.

B. Tests for rare phenomena

In column 144 and in the articles I linked there, I made a similar mistake regarding Munchausen syndrome by proxy, and regarding the interpretation of the results of medical tests for rare diseases. Think of a person who goes to the doctor and the doctor suspects that he has some rare disease, whose rarity in the population is one in a million. He sends the patient for a test that has a reliability of 1% in both directions (that is, one percent of the patients will be found healthy – false positive, and one percent of the healthy will be found sick – false negative). The test finds him sick. He returns to the doctor with the test results. What should the doctor do in such a situation? Remember that this is a very reliable test, since the chance of making a mistake is 1%. In other words: what is the chance that the guy is really sick? It turns out that the chance of this is negligible, and the doctor can send him home happy and good-natured without any worries. Here is the explanation.

Let’s say we have a million people in the population, and we send everyone to take the above test. The prevalence of the disease is 1%, meaning there is only one real patient. How many of the subjects will come out with a positive result? Ten thousand, since the error is 1%, and therefore out of a million healthy people, 10,000 will come out with a positive result as if they were sick. But of those, only one is really sick. Therefore, if the results of the test I took are positive, the chance that I am sick is one in ten thousand. A completely negligible chance. Apparently, the result of my test is completely clear, since the chance of error is 1%, but it is still a complete coincidence. Note that this is exactly the same phenomenon as the black swan we encountered above. A result that seems clear, but when you look more closely, you discover that it is completely coincidental and no conclusion can be drawn from it. If there is a disease whose prevalence is one in a million, then there is no point in conducting a test for it that is 1% reliable. A test that is 1 in a million reliable begins to be indicative. (The chance of error is still 50%). The reliability of the test should be of the order of magnitude of the prevalence of the disease. I like to describe it as a net designed to catch fish, where it is clear that the size of the holes in the net (the chance of error) should be less than the size of the fish (the prevalence of the disease).

Alternatively, if there is some other indication, however weak, that I am sick (which is usually the case if the doctor sends me for the test), that of course completely changes the situation. Why? Because within the group of those who have that indication the prevalence of the disease is already much higher, and therefore a test with a reliability of 1% is already good enough. If we assume that one in ten who have this symptom is sick, which in itself is a pretty weak indication, but that makes a test with a reliability of 1% a good enough test (10% chance of error).

The same is true for legal evidence. Suppose a person is accused of murder, and evidence is presented against him with 99% reliability. What is the chance that he is really the murderer? Absolutely negligible. The percentage of murderers in the population is zero, so to convict a murderer you need evidence with a reliability significantly higher than this prevalence (i.e., small enough holes in the evidentiary network). But when someone is brought to trial, it is done because there is some indication that he is the murderer, for example that he had the opportunity (he was around). Out of ten people who were around, one is the murderer, meaning that in this group the prevalence is 10%, so 99% evidence is already approaching good enough.

I once explained the legal rule that a person is convicted by his own confession only if there is an addition of ‘something’ against him. What is the logic in this rule? I assume that lawyers don’t really know how to explain it (just as many doctors aren’t really aware of the necessary relationship between the reliability of the test and the prevalence of the disease. In recent years, this has already been taught in medical schools), but I think this is the explanation. The chance that a person will lie and simply convict himself is zero. But the percentage of criminals is also zero, and therefore a confession is not good enough evidence. But if there is something else that constitutes an indication that it is indeed a crime, it greatly reduces the group and increases the prevalence, and now good evidence such as self-incrimination becomes reliable enough.

C. Munchausen syndrome by proxy

In my article in Asia , “The Failure of Representation in Halacha,” I dealt with this phenomenon, surrounding Munchausen syndrome by proxy. Ms. Sally Clarke is a British woman whose two babies died in her home from unexplained deaths (cot death). She was charged in a British court with the murder of her children, and was convicted and sentenced to prison. The conviction was based on expert testimony by Professor Sir Roy Meadow, who claimed that the chance of cot death is: 1/8,500. Therefore, the chance of two children dying is the square of this small number, which comes out to approximately 1/73,000,000. Professor Meadow claimed that there is a medical syndrome called ‘Munchausen syndrome by proxy’ (some have attributed it to the ‘starving mother’ in our country), which means that a person sometimes harms others in order to get attention for himself. He argued that since the chance of SIDS is so small, it is clear that this is murder based on the aforementioned syndrome. Without any supporting evidence, based solely on this statistical consideration, the judge found Sally Clark guilty of murdering her children, and sentenced her to prison. We note that Professor Meadow testified in hundreds of trials, and in many of them the defendants were found guilty and sentenced to various sentences. Some of them without any other supporting evidence (what we called “something” above).

The stupidity of the witness and the judge is unbelievable. Not without reason, after a while, a witness expert in statistics came and testified in court that the conviction was based on a statistical error. His main argument was that it is not correct to multiply the numbers by each other, since the events can be statistically dependent. Even if the chance of a child dying in a crib is 1/8,500, this does not mean that the chance of two dying is the square of this number. Since the causes of crib death are unknown, it is likely that there are factors in the home, or in the genes in the family, that could have caused this death. And since these are two brothers who grew up in the same house, it must be assumed that the cause of their deaths was the same reason, and therefore the events are interdependent. Note that this is really the discussion of the issue of cot deaths. The question of whether these two events are coincidental or have a common explanation is an expression of a general law.

Let’s clarify this with an example. Reuven bought a lottery ticket and his numbers were lucky. What is the chance that these exact numbers will be lucky? Very small (say 1 in a million). And what is the chance that Reuven will win the lottery? It is also very small (say 1 in a million). Now we ask: What is the chance that these numbers will also be lucky and Reuven will also win the lottery? Ostensibly, this is a multiplication, and the result is 1 in a million millions. But this is a mistake, because Reuven’s win is a result of the fact that his numbers were lucky. The events are interdependent, and therefore it is not correct to see in this coincidence something that is more surprising than each of them separately.

If so, the testimony of the statistics expert suggests that the chance is not as small as initially thought. I must say that you don’t have to be a statistics expert to be suspicious of Roy Maddow’s argument. Just a little common sense. By the way, if there were an addition of something that supports the suspicion of murder, that would of course change the picture completely.

But even the statistician witness, if that was indeed his argument (that’s what I read), made a stupid argument. Even if you accept his argument and don’t double the odds, it’s still clear that the chance of two children dying in the crib is very small. So it’s not one in sixty million but one in a million. And isn’t that enough for a criminal conviction? What do I care if you double the odds or not? Furthermore, the chance of one child dying in the crib is 1/8,500, which is also a very small number. According to him, any mother whose child dies in the crib could be sentenced to prison. Such a small chance certainly meets the conditions for a criminal conviction.

The main problem with the testimony of the medical expert (Roy Meadow) above was not the independence of the two deaths as the statistical expert claimed, but a completely different problem. There was an ignoring of the fallacy that I am dealing with here. To see this, we will treat this statistical test as a means of diagnosing murder due to Munchausen syndrome. The reliability of the test is 8,499/8,500, and therefore it supposedly captures murderers with the above syndrome with very high reliability. The problem is that the prevalence of this syndrome is extremely low. How many women in Britain (and indeed the entire world) would murder their sons to get attention? Let’s assume for the sake of discussion that this is a prevalence of about 1/100,000 (which seems like a very large estimate compared to the true prevalence). Now we can immediately see that a test with a chance of error of 1/8,500 is worthless in such a context. The test is very reliable, but its unreliability is much higher than the prevalence of the phenomenon it is trying to capture. The holes in this statistical net are too big for the fish in question.

I’ll take you further. Let’s assume that the odds can be doubled, and the chance of two children dying in a cot is indeed one in sixty million. That is, I assume for the sake of discussion that Roy Madow was right. Does that mean that this mother murdered them? Ostensibly, this is exactly Rabbi’s argument, that after it happened twice, it is assumed that there is a common explanation for both events and that they are not a coincidence. But now it is easy to see that this is a complete mistake. There are several tens of millions of mothers in Britain, so even if the chance of such a murder is one in sixty million, it is expected that there will be one woman in Britain in whose home two children will die by chance from cot death. The fact that we found two such children in her home does not arouse any suspicion, just as if we found a woman whose three husbands had died it does not arouse any suspicion. Broadening our perspective shows us that this clearness is actually pure chance. I assume that out of several tens of millions of mothers in Britain, we would find several thousand homes in which accidental cot death occurred. Of these, there will be one or two cases in which two cot deaths occurred, and again by complete coincidence. Not necessarily due to a common medical cause as the statistical witness claimed (because of which it is forbidden to double the odds, like Rafi Dema or some kind of factor of Avimi Maharonia in the issue of cot deaths brought up in the previous column) [1] and not due to a common criminal cause as Roy Madow claimed (that the mother murdered them). It was simply a case of luck (Rav Ashi there, but in the sense of a case and not of stars). In other words, it is more likely that this is a coincidence, and it is completely predictable if one broadens one’s perspective.

We can see here that these two explanations are two sides of the same coin. Broadening our perspective shows us that looking at small numbers (the deaths of two children in one home) teaches us nothing.

Summary: A Look at Halacha, Empirical Tests, and Statistics

We have seen cases here where very significant findings turn out to be a completely random statistical result. In such cases, the assumption that there is a common explanation and that this is not a coincidence is a statistical error. If I do not have a reasonable explanation a priori, I would prefer to assume that despite the apparent significance, this is actually a random phenomenon. Something happened once, twice, and three times, and yet it is a pure coincidence. Just as we saw with the successful schools with the first letter in their name,

The same should be said about a woman whose husbands have died, or children who have died due to circumcision. In a talkback to the previous column, I commented that this analysis raises a big question mark regarding menstruation. In my opinion, for a woman who gets her period three times on the same day of the month, the chance that it is a natural factor causing this is zero. It is much more logical that it is a coincidence. And certainly if she only got it once and we are just afraid of menstruating this month. This already seems really far-fetched. This is in contrast to a cruise and a mid-season period, which sound more logical.

Although I haven’t checked the facts, before making decisions, one should examine in a statistical survey (i.e., by broadening one’s perspective) the percentage of women who menstruate on a fixed date of the month in relation to the general situation. It is important to be precise here: the question is not how many women menstruated three or four times in a row on the same date of the month, but rather a question of conditional probability: Out of all the women who menstruated three times on the same date, how many menstruated a fourth time? This should be compared to the number of those who menstruate on the same date out of all the women. If the conditional probability is significantly higher, then there is room for this assumption (because it means that there is a correlation between menstruating three times and menstruating a fourth time. When there is no correlation, these two probabilities are equal).

In my opinion, the conditional probability will not be substantially different from the general percentage of women who menstruate on that date. If this is indeed the case, then it is much more reasonable to attribute this recurrence to factors other than the state of the moon and the date in the Hebrew month (as stated above, one can always find an ad hoc explanation that interweaves the three recurrences, and the state of the moon does not seem like the most likely explanation), and perhaps it would be most reasonable to attribute it to pure chance. If indeed the phenomenon is examined by broadening the perspective and I am found to be correct, then it would appear on the surface that this is an error of the Sages (something like the claim that zodiac signs and stars cause things), and in my opinion this halakha will cease to be valid, as is any halakha that is based on a scientific or factual error.

By the way, in response to Tirgitz’s question in the previous column’s talkback , the same is true for a bull that is prone to goring. If we examine the conditional probability of a bull that has gone three times to goring a fourth time (i.e. the next time a suitable candidate for going in front of it appears), and compare it to the probability of an ordinary bull going in by chance, I’m not sure we’ll find a significant difference. If there is indeed no difference, then the proneness of a bull is also actually a wrong law, in which case we are actually talking about a random percentage of bulls that gore three or four times by chance and not because of a goring nature. Alternatively, we can examine the gap between the number of bulls that gore twice and bulls that gore four times, or between four and five. The number of gores four times and five times should be quite similar if it is a goring nature (if we neutralize the chance that the owner has reset and started to take care of his bull properly). All of this requires examination, and its findings will determine the law. Here too, as in all three-fold assumptions, it is a matter of evaluating reality (even though we learn this from a verse. It is the sages who learned it from the verse) and not a binding law that is independent of reality.

[1] It is true that it can be said that cradle names are not really a random event. If it happens to any child, there is probably a reason for it, but we do not know it. Therefore, when the same statistician testifies that there is a connection between the two cases, he actually means to say that there is a natural cause for this death (and it is familial, like Rafi Damma in the issue of the Yevamot Sed that we saw in the previous column) and there is no need to assume that the mother murdered them. This is indeed a correct argument, and it is parallel to mine. But then there is really no point in worrying about doubling the odds and making a correlation between the cases. That is not the main point.