Induction – Continued: On Chance Significance (Column 730)

With God’s help

In the previous column I dealt with generalizations in halakhah and in general; I pointed out the difficulties entailed in processes of generalization and the methods employed to avoid them. We saw the roles played by intuition and a priori reasoning in the matter of generalization. The discussion proceeded through the sugyot that deal with the “three-times” chazakah. One example the Gemara brings for a three-times chazakah is a case in which a woman married three times and on each occasion her husband died immediately after the wedding. In such a case we forbid her to marry again, and according to Rav Ashi (Yevamot 64b)—whose view is accepted as halakhah—this is due to a concern that her mazzal causes her husbands’ deaths, and therefore we fear that her next husband will also die. I explained there that this is a concern rather than a positive conclusion, and that the concern rests on statistically significant findings. The recurrence of an event three times is sufficiently significant to create concern.

One implication of our a priori considerations in the process of generalization is that where we lack a plausible explanation that threads the three cases together, we do not generalize. I further explained that Rav Ashi’s concern that her mazzal caused it is not a shot in the dark. He is not speaking of mazzal in the sense of randomness, but in the sense of astral influence. Following the words of the Mekor Chayim in Hilkhot Pesach cited there, I showed that when there is no explanation, we do not generalize. Thus, for example, if we were to find that the successful schools all have names beginning with a letter between Alef and Het (A–H), we would not infer from this that one should change a school’s name to improve achievement. Why? Because a priori it does not seem to us that a school’s name is relevant to its performance. Likewise, I argued that from our current scientific perspective there are no astral influences; therefore, were I to see a woman whose three husbands died immediately after the wedding (in a manner that clearly rules out her murdering them or their dying due to contact with her), I would not fear that her mazzal caused it and I would not forbid a fourth man to marry her. I would prefer to say that this is a matter of chance (i.e., mazzal in its modern sense of randomness, not astral influence).

This, of course, raises the question whether this is intellectual rigidity. Seemingly there are significant data indicating that mazzal does have an effect (recall that the halakhic assumption is that three recurrences are not chance), and yet I insist on taking as self-evident the scientific assumption that the stars have no such influence on our lives, and I entrench myself in this assumption even against apparently significant statistical data. It seems I refuse to generalize even where a generalization is called for. I claimed there that such samples are accidental; therefore, despite the apparent significance in the data, it is not correct to treat them as a good basis for generalization, since there is no plausible explanation that threads the cases together. A priori reasoning overrides the statistical significance and leads us to the conclusion that nevertheless this is a matter of chance.

In this column I shall try to explain the logic of this approach, and why this is not mere conservatism or mental rigidity. I will propose two perspectives on the phenomenon (we will later see that they are connected): the law of small numbers and broadening the view.

Explanation A: The law of small numbers

There is a statistical phenomenon that deserves attention, related in some way to the law of small numbers—even when the numbers are not all that small—from which it becomes clear that even an apparently significant phenomenon may be due to chance.

Consider the example we saw in the previous column, where the highest-achieving schools were small. We saw that it is dangerous to infer from this that a small school helps improve achievement. There I attributed this to a statistical error: the analysis did not look at the lowest-achieving schools (which were also small). We saw that the two tails of the Gaussian will contain the cases where the numbers are small (small schools or small towns in the kidney-disease example).

But what if we really were to see that the ten most successful schools are small, and the unsuccessful ones are not small? Should we then infer that a small school helps improve achievement? There is a logic to that—and yet it is important to understand that there is no situation in which I could not find some rule that threads together all the successful schools. For example, that each one appears in an odd position on the list of schools in the country; or that the first letter of its name is between Alef and Het; or that they all have fences made of non-wood material, a principal taller than 1.80 m, and so on. There will be no list of schools—or of towns with low morbidity—that I could not explain ad hoc using some arbitrary parameter. Such a parameter will always be found.

In the previous column (in the section “The difficulty in this conclusion”) I mentioned column 482, where I dealt with this phenomenon in general; here I shall elaborate a bit more. Consider the sequence of numbers …3, 5, 7. What is the next number? Seemingly 9, since this appears to be the sequence of odd numbers. But I could just as well propose 11 and explain that this is the sequence of primes (numbers divisible only by themselves and by 1; 9 is not prime because it is divisible by 3). So who is right? No one—both explanations are possible. In that column I showed that this is true for any sequence you choose: one can find countless rules that explain it, and each rule dictates a different next number. This is exactly like my remark regarding the list of high-achieving schools. One can attribute the list to their being small, or to their having a wooden fence, or to the principal being over 1.80 m—countless explanatory options, all of which fit. I assume that we will nevertheless choose the explanation that they are all small, because we have a good story behind it. It seems to us that this parameter—unlike the others—is relevant.

Thus, a priori reasoning dictates whether and how we generalize. In column 715 we saw this from another angle, and I explained that without initial intuitions no scientific generalization is possible. The upshot is that although Kahneman is right in saying that the presence of a good story can bias us toward erroneous conclusions, at the same time such a story is very important when we wish to choose among several possible generalizations. As we saw in the previous column, our a priori sense of the relevance of the explanatory parameter to the phenomena explained is crucial. Without this a priori elimination we will always have countless explanations for any finding. The error Kahneman points to is that sometimes we choose a good story when there is another story no less good, or that sometimes the story biases our statistical analysis (as in the school-achievement and morbidity examples). This, of course, must not happen. But given several possible generalizations, it will always be the story that seems to us a priori the best that will dictate the direction of the generalization.

And what if we have no good story? That is, if a priori it is clear to us that the examples we observed share no connecting thread and that there is no generalization that explains them? In such a case, indeed, we will sometimes give up 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 explaining schools’ achievements by the first letter of their names, I prefer the interpretation that this is coincidental rather than attributing it to mazzal (astral influence) or other bizarre factors for which I have no indication. This is not rigidity but rational elimination that does rest on our a priori assumptions. We should remain open to the possibility that those assumptions are wrong, but so long as this has not been demonstrated convincingly, I would not dismiss them.

Explanation B: Broadening the view

Let us consider this from another angle. The fact is that different people die in different ways and for various—and sometimes unknown—reasons (sometimes known, sometimes not). In a survey here you can see that some halakhic authorities discuss the husband’s death at any point after the wedding (i.e., even if he died of old age fifty years later), but this truly seems entirely implausible. For simplicity, then, let us speak of death within the first year of marriage. This can, of course, occur completely by chance.

Assume, then, for the sake of discussion, that we find statistically that in one out of a hundred married couples the man dies in the year after the wedding. In each such case there remains a widow. That is, about one percent of the women in the population are those whose husbands died in the first year. Now one of them comes and wishes to remarry. Is there any logic in forbidding it to her? Clearly not. The death of one husband certainly does not serve as an indication to suspect that the woman’s mazzal caused the death, and there is no logic in forbidding another man to marry her. Even without any special mazzal of hers there are natural deaths of men, and it was (in the modern sense) her “luck” that such a man happened to be her spouse. Suppose we have a million married women in the country; ten thousand of them will be women whose husbands died in the first year of marriage. If so, when we see a woman whose husband died, is it reasonable to fear that her mazzal caused his death and to forbid her to remarry? Clearly not. This is simply the statistics of male mortality, unrelated to the woman’s mazzal. Note that although the chance that such a husband will die is only 1%, it is clear that the chance that the woman’s mazzal caused the death is negligible. Even without any astral influence there are ten thousand such women, and this woman is one of them. This is mazzal in the modern sense (randomness), and there is no reason to posit the influence of the stars.

What about a woman whose two husbands died? If her mazzal caused it, this would mean there is a common cause for the deaths of her two husbands (the woman’s mazzal). But note that even if there is no mazzal effect at all and we are looking only at the statistics of male deaths in the year after marriage, there will still be about a hundred women in the population whose two husbands will die in the first year. Therefore, if we meet a woman whose two husbands have died, there is no reason to think her mazzal caused it. She is one of the hundred women we expect to be in such a situation (out of a million women in the country), and thus encountering such a woman should not arouse any fear or suspicion that she carries lethal mazzal. This is a statistically expected outcome, unrelated to stars and mazzal.

If we were to see that in some locale the number of women “who cause husbands to die” is much higher than the global baseline, that would be a different story—and even then we would need to run regressions to rule out chance (a small locale can yield such results) or identify some other cause. Alternatively, if a single woman were to have ten husbands die within the first year, and upon checking around we see that there is no woman with eight dead husbands, or seven, or six, or five—then there would be room for suspicion (though even there I would prefer to suspect that she killed them rather than that her mazzal caused it). These are significant statistical indications that can justify generalization. But under an ordinary distribution of deaths it is clear that there will be not a few women for whom two husbands die, and therefore when one of them appears before us there is no reason to fear anything. This tells us nothing about the fate of the next husband of such a woman. Rabbi’s view—fearing after two husbands’ deaths—is highly implausible.

What if three of the woman’s husbands were to die? It seems that here we are at the statistical boundary, since if we are dealing with a random distribution of male deaths, there is only a one-in-a-million chance that three husbands would die for the same woman during the first year after marriage. That is, only for one woman out of the million women in the country could this happen—only one. This already provides some basis for suspicion, and there one should begin to be concerned (though not to decide positively, of course). In such a case, even without any evidence in the three cases, even Rabban Shimon ben Gamliel would already warn the fourth woman not to marry this dangerous fellow. You can now understand the basis of the dispute between Rabbi (who establishes a chazakah after two cases) and Rabban Shimon ben Gamliel (who holds that it is established after three). The question is: from when does a reasonable suspicion begin? Clearly there is no sharp statistical line.

But I think that with the numbers I have given, even if three husbands died I still would not be concerned. At least, if indeed there are ten thousand women with one dead husband and a hundred with two dead husbands, then when I find one with three dead husbands there is no reason to assume this is not the outcome of the expected random distribution of deaths. Here too I would not be concerned. If there were no women with dead husbands in the first year at all, and there was only one with three dead husbands—all in the first year—then we should indeed be suspicious.

This is the phenomenon of broadening the view. When we broaden our view beyond the case before us and examine the distribution in the population as a whole, we discover that the case which at first glance looked significant is merely a statistically expected instance. This explains the conclusion at the end of the previous column: where there is no reason to suspect a common explanation behind all the recurrences, it is more reasonable to assume these are mere coincidences. This is not stubbornness but rational thinking.

A real-life example: Shimon Cooper and his three wives

My good friend, retired judge Menachem Finkelstein, chaired a panel that dealt with a similar case. A book was even written about it, To Catch a Murderer, by Omri Assenheim, who also did a television investigation of the affair (and thereby had some part in the conviction). A man named Shimon Cooper was convicted in 2016 of the murder of his first wife (the murder occurred in 1994) and of his third wife (in 2009). Initially, the case regarding the third wife was closed for lack of evidence, but new evidence emerged and he was convicted. Part of the conviction for the first wife’s murder rested on the fact that he was convicted of the third wife’s murder (both were perfectly healthy and died suddenly). That is practically a chazakah—though here it came after a single instance. Incidentally, Cooper was later investigated on suspicion of murdering another woman, a Holocaust survivor he was suspected of trying to exploit.

What would have happened had a third wife of his died? I do not know whether two such murders would suffice to convict him of a third. Seemingly, according to Rabbi there is room for it, since Rabbi establishes a chazakah after two occurrences. But it is quite clear that this is not right. Rabbi is speaking of suspicion and concern, not of a positive conclusion. That is, Rabbi would warn a woman who wished to marry this Cooper and would forbid her to do so because of the chazakah. But a criminal conviction requires a positive conclusion. In conversations I had with Finkelstein at the time we discussed such statistical phenomena at length. I will now bring several of the examples I raised with him then.

A. The Black Swan

Nassim Taleb, in his book The Black Swan, pointed to a similar phenomenon. Consider an investor with exceptional, long-term success, such as Warren Buffett. I assume you would agree that he has fantastic abilities and understanding in financial investing and in assessing companies’ prospects. Surprisingly, Taleb challenges this conclusion, with the following argument.

Imagine a market composed entirely of investors who have no understanding or ability whatsoever, each choosing investments entirely at random. For the sake of discussion, suppose that out of ten such investors, one will show consistent profit for an entire month; out of a hundred, one will profit consistently for a year; out of a thousand, one will profit consistently for ten years; out of ten thousand, one for thirty years; and out of a million, one will profit consistently for a hundred years. Suppose, for the sake of discussion, that Warren Buffett has profited consistently for fifty years. Does that necessarily mean he has exceptional talent and insight? Not necessarily. It is possible that he is just a random investor—and since there are millions of investors in the world, statistically one of them should profit consistently for a hundred straight years even without talent. If so, Buffett might be entirely without special talent, and his profits might derive from the fact that he happens to be that one.

This is, again, the phenomenon of broadening the view. Here, too, when we broaden the view beyond the case before us and examine the distribution across the population, we discover that a case that at first glance looks significant may be entirely accidental, a consequence of simple statistical calculation. True, if we were to broaden the view and discover that Buffett is particularly outstanding—that there are no investors with forty, thirty, or twenty years of success, or that he continues to profit for more and more years while all other players around him drop off the board—then perhaps it would be appropriate to infer that his success indicates exceptional ability. But to reach such a conclusion we must broaden the view. From a narrow view focused solely on the case at hand, no conclusion can be drawn. Although there appears to be impressive significance, it may be mere chance.

Incidentally, even this conclusion can be challenged. It may be that even if we find the pyramid of success I described, what produces it is the distribution of investors’ talents (rather than a distribution of accidental successes unrelated to talent). Therefore, the statistics of investor success do not necessarily say what Taleb claims. But the possibility he raises truly exists, and that is what matters for our purposes. It obliges us to think carefully before drawing conclusions from a chain of events that looks significant. So too with “three-times” chazakot: there is a possibility that despite the apparent recurrence, we are dealing with chance.

B. Tests for rare phenomena

In column 144 and in the article linked there, I discussed a similar error concerning Munchausen syndrome by proxy and the interpretation of the results of medical tests for rare diseases. Consider a person who comes to a physician, and the physician suspects he has some rare disease whose prevalence in the population is one in a million. He sends the patient for a test whose error rate is 1% in both directions (i.e., one percent of the sick will be found healthy—false negatives—and one percent of the healthy will be found sick—false positives). The test returns positive. The patient returns to the doctor with the results—what should the doctor do? Remember, this is a very reliable test: the chance of error is 1%. In other words: what is the probability that the person is truly ill? It turns out that the probability is negligible, and the doctor can send him home happy and unworried. Here is the explanation.

Suppose we have a population of one million people, and we send them all to take the test. The disease’s prevalence is one in a million, i.e., there is only one true patient. How many will come out positive? Ten thousand—because the false-positive rate is 1%, and therefore out of a million healthy people ten thousand will test positive as if they were sick. But of those only one is truly sick. Therefore, if my test is positive, the probability that I am sick is one in ten thousand—a negligible probability. Seemingly, my test result is highly significant, since the chance of error is 1%, and yet this is still mere chance. Note that this is the exact same phenomenon as the Black Swan above. A result that appears significant, but when we broaden the view we discover it is entirely accidental and no conclusion should be drawn from it. If a disease’s prevalence is one in a million, there is no point in testing for it with a test whose error rate is 1%. Only a test with an error rate of about one in a million would begin to be indicative (and even then the error probability is 50%). The test’s accuracy needs to be of the same order of magnitude as the disease’s prevalence. I often describe this as a net designed to catch fish: clearly, the holes in the net (error rate) must be smaller than the fish (prevalence) you intend to catch.

Alternatively, if there is some other indication, however weak, that I am ill (and this is usually the case when a doctor sends someone for testing), the picture changes entirely. Why? Because within the group that has that indication, the disease’s prevalence is much higher, and therefore a test with a 1% error rate is already good enough. If, say, one out of ten with that symptom is ill—a rather weak indication in itself—this turns a 1%-error test into a sufficiently informative one (about a 10% chance of error).

The same applies to legal evidence. Suppose a person is accused of murder, and a piece of evidence with 99% reliability is brought against him. What is the likelihood that he is truly the murderer? Negligible. The percentage of murderers in the population is minuscule; to convict a murderer one needs evidence with reliability far higher than that base rate (that is, the holes in the evidentiary net must be much smaller). But when someone is put on trial, it is because there is some indication that he is the murderer—e.g., that he had the opportunity (he was nearby). Among ten people who were nearby, one is the murderer; i.e., within that group the base rate is 10%, and evidence at 99% reliability already approaches being sufficiently probative.

I once explained the legal rule that a person may be convicted on his own confession only if there is an additional “something” (davar mah) against him. What is the logic of this rule? I assume jurists do not really know how to explain it (just as many physicians are not truly aware of the relationship required between a test’s reliability and a disease’s prevalence—though in recent years this is taught in medical schools). I think this is the explanation: the chance that a person would lie and falsely incriminate himself is tiny. But the percentage of criminals is also tiny; therefore, a confession is not sufficient evidence. However, if there is an additional “something” that indicates he indeed committed the crime, that greatly shrinks the relevant group and increases the prevalence; now strong evidence like self-incrimination becomes sufficiently reliable.

C. Munchausen syndrome by proxy

In my article in Asya, “The Representativeness Fallacy in Halakhah,” I discussed this phenomenon around Munchausen syndrome by proxy. Mrs. Sally Clark was a British woman whose two infants died at home of unexplained causes (SIDS). She was charged in a British court with murdering her children, convicted, and imprisoned. The conviction was based on expert testimony by Prof. Sir Roy Meadow, who claimed that the probability of SIDS is 1/8,500. Therefore, the probability of two such deaths is the square of that small number, about 1/73,000,000. Prof. Meadow argued that there is a medical syndrome known as “Munchausen by proxy” (sometimes likened here to the “starving mother”), meaning that a person may harm others to gain attention. He argued that since the probability of SIDS is so small, it is clear that we are dealing with murder due to that syndrome. Without any corroborating evidence—solely on the basis of this statistical consideration—the judge found Sally Clark guilty of murdering her children and sentenced her to prison. Note that Prof. Meadow testified in hundreds of trials, and in many of them defendants were found guilty and sentenced—some without any other corroborating evidence (what we called above “davar mah”).

The stupidity of the expert and the judge is unbelievable. Not for nothing, later a statistics expert testified that the conviction was based on a statistical error. His main claim was that it is incorrect to multiply the probabilities, since the events may be statistically dependent. Even if the probability of SIDS for a child is 1/8,500, it does not follow that the probability for two children is the square of that number. Since the causes of SIDS are unknown, it is reasonable to assume there are factors in the home, or in the family’s genes, that might cause such death. And since we are dealing with two siblings who grew up in the same home, the cause of death was likely the same; hence, the events are dependent. Note that this is precisely the discussion in Yevamot 64: are these two accidental events, or do they share a common cause—an expression of a general law?

Let us clarify with an example. Reuven bought a lottery ticket and his numbers won. What is the probability that those very numbers would be drawn? Very small (say, one in a million). And what is the probability that Reuven would win the lottery? Also very small (say, one in a million). Now ask: what is the probability that both those numbers would be drawn and Reuven would win the lottery? Seemingly we multiply and get one in a trillion. But that is a mistake, because Reuven’s winning is a consequence of his numbers being drawn. The events are dependent; therefore, it is incorrect to treat their conjunction as more surprising than each alone.

Thus, from the statistician’s testimony it follows that the probability is not as small as initially thought. I must say: one need not be a statistics expert to be suspicious of Meadow’s argument—just a bit of common sense. Incidentally, were there additional evidence supporting suspicion of murder, that would, of course, change the picture entirely.

But the statistician too, if that was indeed his argument (as I read), advanced a foolish claim. Even if we accept his point and do not multiply the probabilities, it remains clear that the probability of two SIDS deaths is very small. So it is not one in sixty million but one in a million. Is that not sufficient for a criminal conviction? What do I care whether we multiply or not? Moreover, the probability of a single SIDS death is 1/8,500, which is also very small. On his view, any mother whose child died of SIDS could be imprisoned. Such a small probability would certainly meet the standards of criminal conviction.

The main problem with the medical expert’s (Meadow’s) testimony was not the independence of the two deaths, as the statistician claimed, but an entirely different problem: ignoring the very fallacy I am discussing here. To see this, treat the statistical “test” as a diagnostic for murder due to Munchausen by proxy. The test’s “reliability” is 8,499/8,500; seemingly it captures the murderers with that syndrome with very high accuracy. The problem is that the prevalence of that syndrome is extremely low. How many women in Britain (or anywhere) murder their children to gain attention? Let us assume, generously, a prevalence of about 1/100,000 (surely an overestimate). Now it is immediately clear that a “test” with a 1/8,500 error rate is worthless here. The test is very accurate, but its inaccuracy is far higher than the prevalence of the phenomenon it seeks to capture. The holes in this statistical net are larger than the fish.

I will go further. Suppose we may multiply the probabilities, and the probability of two SIDS deaths is indeed one in sixty million—i.e., suppose Meadow was right. Does that mean the mother murdered them? Seemingly, that is precisely Rabbi’s argument: after something happens twice, there is a chazakah that there is a common cause and it is not mere chance. But now it is easy to see that this is a complete mistake. There are tens of millions of mothers in Britain; even if the probability of such a coincidence is one in sixty million, we still expect that there will be one mother in Britain in whose home two SIDS deaths occur by chance. The fact that we found such a home raises no suspicion—just as finding a woman with three dead husbands raises no suspicion. Broadening the view shows us that this “significance” is actually pure chance. I assume that out of the many millions of mothers in Britain, we would find several thousand homes where a single SIDS death occurred by chance; among them there will be one or two homes where two such deaths occurred—again, purely by chance. Not necessarily due to a shared medical cause, as the statistician claimed (which would bar multiplying the probabilities, akin to “Rafi Dama” or a factor like “Avimi of Hagrunia” in Yevamot 64 cited in the previous column)[1], and not due to a shared criminal cause, as Meadow claimed (that the mother murdered them). Simply: “her mazzal caused it” (Rav Ashi)—but in the sense of chance, not stars. That is, it is more reasonable that this is mere coincidence, which is entirely expected once we broaden the view.

Here we can see that these two explanations are two sides of the same coin. Broadening the view shows that looking at small numbers (two deaths in one home) teaches nothing.

Summary: A view on halakhah, empirical tests, and statistics

We have seen cases in which highly “significant” findings turn out to be entirely accidental statistical results. In such cases, the assumption that there is a common cause and that this is not chance is a statistical error. If I lack a plausible a priori explanation, I will prefer to assume that despite the apparent significance, the phenomenon is accidental. Something happened once, twice, three times—and nevertheless it is mere chance. Just as we saw with successful schools whose names start with a given letter,

so too we should rule regarding a woman whose several husbands died, or infants who died due to circumcision. In a comment to the previous column I noted that this analysis raises a major question regarding vesetot (menstrual cycles). In my estimation, if a woman gets her period three times on date V of the month, the probability that there is some natural factor causing this is negligible; it is far more reasonable that this is chance. And certainly if she got it only once and we merely fear a monthly veset—that already seems outright implausible. This is unlike a veset ha-haflagah (interval-based) and onah beinonit (“average onah”), which sound more reasonable.

I have not examined the matter empirically, and before making decisions one should survey it statistically (i.e., by broadening the view): what percentage of women get their period on a fixed date in the month relative to the general situation. Precision is important here: the question is not how many women got their period three or four consecutive times on the same date in the month, but a question of conditional probability: out of all women who got it three times on the same date, how many got it also the fourth time? This should be compared with the percentage of those who get it on that date out of all women. If the conditional probability is significantly higher, there is room for such a chazakah (because that would indicate a correlation between three occurrences and the fourth; when there is no correlation, the two probabilities are equal).

My estimate is that the conditional probability will not differ materially from the overall percentage of women who get their period on that date. If this is indeed the case, then it is far more reasonable to attribute the recurrence to other factors rather than to the lunar phase or the date in the Hebrew month (as noted above, one can always find an ad hoc explanation that threads the three occurrences, and the lunar phase does not seem the most plausible explanation). Perhaps the most reasonable explanation would be simple chance. If, upon broadening the view, the phenomenon is examined and my assessment is borne out, it would seem that this is an error of the Sages (akin to the claim that stars and constellations cause things), and in my view such a halakhah would cease to be valid, like any halakhah based on a scientific or factual error.

Incidentally, in answer to Tirgitz’s question in a comment on the previous column, the same applies to the “mu’ad ox.” If we examine the conditional probability that an ox which gored three times will gore a fourth time (i.e., the next time a worthy target appears before it), and compare it with the probability of a random ox goring, I am not sure we will find a significant difference. If indeed there is no difference, then the status of mu’ad would be a mistaken ruling; in such a situation we are merely seeing a random percentage of oxen that gored three or four times by chance and not due to a goring nature. Alternatively, we could examine the gap between the number of oxen that gored twice and those that gored four times, or between four and five. The numbers of those that gored four times and five times should be fairly similar if we are dealing with a goring nature (setting aside the possibility that the owner woke up and began guarding his ox properly). All this requires examination, and the findings should determine the ruling. Here too—as in all “three-times” chazakot—we are dealing with an assessment of reality (despite the fact that a verse is cited; the Sages are the ones who derived it from the verse), not a binding rule independent of reality.

[1] One could say that SIDS is not truly a random event. If it occurs to a given infant, it is likely there is a cause—we just do not know it. If so, when that statistician testifies that there is a connection between the two cases, he basically means there is a natural cause for the deaths (and it is familial, like “Rafi Dama” in Yevamot 64 as we saw in the previous column), and there is no need to assume the mother murdered them. That argument is indeed correct and parallels mine. But then there is no point in fixating on the multiplication of probabilities and on the dependence between the cases; that is not the main point.