Ketuvot 15a ~ Talmudic Probability Theory

תלמוד בבלי כתובות דף טו עמוד א 

  א"ר זירא: כל קבוע כמחצה על מחצה דמי ..מנא ליה לר' זירא הא? ...מתשע חנויות, כולן מוכרות בשר שחוטה ואחת בשר נבלה, ולקח מאחת מהן ואינו יודע מאיזה מהן לקח - ספיקו אסור, ובנמצא - הלך אחר הרוב, 

R. Zera said: Any doubt about something that is fixed in its place is considered be a fify-fifty chance... Where does he learn this from ? [From a Baraisa which teaches the following. Consider a town in which] there are nine shops, all of which sell kosher meat, and one store that sells meat that is not kosher. If a person bought meat from one of these [ten] stores but he cannot recall from which, his doubt means that the meat is forbidden. But if he found a piece of meat [in the street and he cannot tell from which store it came] he may follow the majority [and assume the meat is kosher]...

As Dov Gabbay and Moshe Koppel noted in their 2011 paper, there is something odd about talmudic probability. If we find some meat in an area where there are p kosher stores and q non-kosher stores, then all other things being equal, the meat is kosher if and only if p > q.This is clear from the parallel text in Hullin (11a) where the underlying principle is described as זיל בתר רובא – follow the majority. Or as Gabbay and Koppel explain it:

“Given a set of objects the majority of which have the property P and the rest of which have the property not-P, we may, under certain circumstances, regard the set itself and/or any object in the set as having property P.
”

— Gabbay and Koppel 2010

In other words, what happens is that if there are more kosher stores than there are treif, the meat is considered to have become kosher. It's not that the meat is most likely to be kosher and may therefore be eaten.  Rather it takes on the property of being kosher. 

We encountered another example of talmudic probability theory only a week ago, on Ketuvot 9a. There, a newly-wed husband claims that his wife was not a virgin on her wedding night. The Talmud argues that his claim needs to be set into a context of probabilities:

1. She was raped before her betrothal.

2. She was raped after her betrothal.

3. She had intercourse of her own free will before her betrothal.

4. She had intercourse of her own free will after her betrothal.

Since it is only the last of these that renders her forbidden to her husband (stay focussed and don't raise the question of a husband who is a Cohen), the husband's claim is not supported, based on the probabilities. Here is how Gabbay and Koppel explain the case - using formal logic:

Oh, and the reference to Bertrand's paradox? That is the paradox in which some questions about probability - even ones that seem to be entirely mathematical, have more than one correct solution; it all depends on how you think about the answer. One if its formulations goes like this: Given a circle, find the probability that a chord chosen at random will be longer than the side of an inscribed equilateral triangle. Turns out there are three correct solutions. Gabbay and Koppel claim that just like that paradox, the solution to many talmudic questions of probability will have more than one correct answer, depending on how you think about that answer.

Rabbi Nahum Eliezer Rabinovitch (1928-2020) was the Rosh Yeshiva of the hesder Yeshivah Birkat Moshe in Ma'ale Adumim.  (He also had a PhD in the Philosophy of Science from the University of Toronto, published in 1973 as Probability and Statistical Inference in Ancient and Medieval Jewish Literature.)  Rabbi Rabinovitch seemed to have been the first to point out the relationship between Bertrand's paradox and talmudic probability theory in his 1970 Biometrika paper Combinations and Probability in Rabbinic Literature. There, the Rosh Yeshiva wrote that "the rabbis had some awareness of the different conceptions of probability as a measure of relative frequencies or a state of general ignorance."

James Franklin, in his book on the history of probability theory, notes that codes like the Talmud (and the Roman Digest that was developed under Justinian around 533) "provide examples of how to evaluate evidence in cases of doubt and conflict.  By and large, they do so reasonably. But they are almost entirely devoid of discussion on the principles on which they are operating." But it is unfair to expect the Talmud to have developed a notion of probability theory as we have it today. That wasn't its interest or focus. Others have picked up this task, and have explained the statistics that is the foundation of  talmudic probability. For this, we have many to thank, including the late Rosh Yeshiva, Rabbi Rabinovitch.

“(The [Roman] Digest and) the Talmud are huge storehouses of concepts, and to be required to have an even sketchy idea of them is a powerful stimulus to learning abstractions.”

— James Franklin. The Science of Conjecture: Evidence and Probability Before Pascal, 349.

Pesachim 9b ~ Talmudic Probability Theory

Our new tractate Pesachim, deals with all things Paschal. (Well, nearly all). What happens if there were nine piles of matzah and one pile of forbidden leavened bread known as chametz, and along came a mouse and took a piece from one of the piles and carried it into a house that had already been searched for chametz. Must the house be searched a second time? To find an answer, the Talmud quotes a Baraiasa that deals with an analogous question.

פסחים ט, ב 

דִּתְנַן: תֵּשַׁע חֲנוּיוֹת, כּוּלָּן מוֹכְרִין בְּשַׂר שְׁחוּטָה, וְאַחַת מוֹכֶרֶת בְּשַׂר נְבֵלָה, וְלָקַח מֵאַחַת מֵהֶן, וְאֵינוֹ יוֹדֵעַ מֵאֵיזֶה מֵהֶן לָקַח — סְפֵיקוֹ אָסוּר.

With regard to nine stores in a city, all of which sell kosher meat from a slaughtered animal, and one other store that sells meat from unslaughtered animal carcasses, and a person took meat from one of them and he does not know from which one he took the meat, in this case of uncertainty, the meat is prohibited.

וּבַנִּמְצָא — הַלֵּךְ אַחַר הָרוֹב

And in the case of meat found outside, follow the majority.

What this boils down to is this. If most stores in the city sell kosher meat then a piece of meat that is found in the city (that is “outside”) is assumed to be kosher, since the majority of the stores sell only kosher meat. But if a person bought meat from one of the ten stores, but he cannot recall whether or not it was from a kosher store, the meat may not be eaten. In this latter case, we assume that there were simply an equal number of kosher and non-kosher stores. There is a 50-50 chance that the meat comes from a non-kosher store, and it may not be eaten.

By analogy, if the mouse took the morsel from one of the piles, the legal status of the morsel is that of an equally balanced uncertainty concerning whether it was taken from a pile of matzah or a pile of chametz. Consequently, the owner is required to go back and search the house all over again.

Talmudic Probability

As Dov Gabbay and Moshe Koppel noted in their 2011 paper, there is something odd about talmudic probability. If we find some meat in an area where there are p kosher stores and q non-kosher stores, then all other things being equal, the meat is kosher if and only if p > q.This is clear from the parallel text in Hullin (11a) where the underlying principal is described as זיל בתר רובא – follow the majority. Or as Gabbay and Koppel explain it:

Given a set of objects the majority of which have the property P and the rest of which have the property not-P, we may, under certain circumstances, regard the set itself and/or any object in the set as having property P.

In other words, what happens is that if there are more kosher stores than there are non-kosher, the meat is considered to have become kosher. It's not that the meat is most likely to be kosher and may therefore be eaten.  Rather it takes on the property of being kosher. 

We encountered another example of talmudic probability theory when we studied the tractate Ketuvot. There, a newly-wed husband claims that his wife was not a virgin on her wedding night. The Talmud argues that his claim needs to be set into a context of probabilities:

1. She was raped before her betrothal.

2. She was raped after her betrothal.

3. She had intercourse of her own free will before her betrothal.

4. She had intercourse of her own free will after her betrothal.

Since it is only the last of these that renders her forbidden to her husband (stay focussed and don't raise the question of a husband who is a Cohen), the husband's claim is not supported, based on the probabilities. Here is how Gubbay and Koppel explain the case - using formal logic:

Oh, and the reference to Bertrand's paradox? That is the paradox in which some questions about probability - even ones that seem to be entirely mathematical, have more than one correct solution; it all depends on how you think about the answer. One if its formulations goes like this: Given a circle, find the probability that a chord chosen at random will be longer than the side of an inscribed equilateral triangle. Turns out there are three correct solutions. Gubbay and Koppel claim that just like that paradox, the solution to many talmudic questions of probability will have more than one correct answer, depending on how you think about that answer.

Rabbi Nahum Eliezer Rabinovitch, who died in May of this year at the age of 92 was the Rosh Yeshiva of the hesder Yeshivah Birkat Moshe in Ma'ale Adumim.  (He also had a PhD. in the Philosophy of Science from the University of Toronto, published in 1973 as Probability and Statistical Inference in Ancient and Medieval Jewish Literature.)  Rabbi Rabinovitch seems to have been the first to point out the relationship between Bertrand's paradox and talmudic probability theory in his 1970 Biometrika paper Combinations and Probability in Rabbinic Literature. There, the Rosh Yeshiva wrote that "the rabbis had some awareness of the different conceptions of probability as a measure of relative frequencies or a state of general ignorance."

James Franklin, in his book on the history of probability theory, notes that codes like the Talmud (and the Roman Digest that was developed under Justine c.533) "provide examples of how to evaluate evidence in cases of doubt and conflict.  By and large, they do so reasonably. But they are almost entirely devoid of discussion on the principles on which they are operating." But it is unfair to expect the Talmud to have developed a notion of probability theory as we have it today. That wasn't its interest or focus. Others have picked up this task, and have explained the statistics that is the foundation of  talmudic probability. For this, we have many to thank, including the late mathematician and Rosh Yeshiva, Rabbi Rabinovitch.

“(The [Roman] Digest and) the Talmud are huge storehouses of concepts, and to be required to have an even sketchy idea of them is a powerful stimulus to learning abstractions.”

— James Franklin. The Science of Conjecture: Evidence and Probability Before Pascal, 349.

Niddah 18a ~ Talmudic Probability Theory

נדה יח, א 

  א"ר זירא: כל קבוע כמחצה על מחצה דמי ..מנא ליה לר' זירא הא? ...מתשע חנויות, כולן מוכרות בשר שחוטה ואחת בשר נבלה, ולקח מאחת מהן ואינו יודע מאיזה מהן לקח - ספיקו אסור, ובנמצא - הלך אחר הרוב, 

R. Zera said: Any doubt about something that is fixed in its place is considered be a fify-fifty chance... Where does he learn this from ? [From a Baraisa which teaches the following. Consider a town in which] there are nine shops, all of which sell kosher meat, and one store that sells sells meat that is not kosher. If a person bought meat from one of these [ten] stores but he cannot recall from which, his doubt means that the meat is forbidden. But if he found a piece of meat [in the street and he cannot tell from which store it came] he may follow the majority [and assume the meat is kosher]...

As Dov Gabbay and Moshe Koppel noted in their 2011 paper, there is something odd about talmudic probability. If we find some meat in an area where there are p kosher stores and q non-kosher stores, then all other things being equal, the meat is kosher if and only if p > q.This is clear from the parallel text in Hullin (11a) where the underlying principal is described as זיל בתר רובא – follow the majority. Or as Gabbay and Koppel explain it:

“Given a set of objects the majority of which have the property P and the rest of which have the property not-P, we may, under certain circumstances, regard the set itself and/or any object in the set as having property P.
”

— Gabbay and Koppel 2010

In other words, what happens is that if there are more kosher stores than there are trief, the meat is considered to have become kosher. It's not that the meat is most likely to be kosher and may therefore be eaten.  Rather it takes on the property of being kosher. 

We encountered another example of talmudic probability theory way back in February 2015 on page 9a of Ketuvot. There, a newly-wed husband claims that his wife was not a virgin on her wedding night. The Talmud argues that his claim needs to be set into a context of probabilities:

1. She was raped before her betrothal.

2. She was raped after her betrothal.

3. She had intercourse of her own free will before her betrothal.

4. She had intercourse of her own free will after her betrothal.

Since it is only the last of these that renders her forbidden to her husband (stay focussed and don't raise the question of a husband who is a Cohen), the husband's claim is not supported, based on the probabilities. Here is how Gubbay and Koppel explain the case - using formal logic:

Oh, and the reference to Bertrand's paradox? That is the paradox in which some questions about probability - even ones that seem to be entirely mathematical, have more than one correct solution; it all depends on how you think about the answer. One if its formulations goes like this: Given a circle, find the probability that a chord chosen at random will be longer than the side of an inscribed equilateral triangle. Turns out there are three correct solutions. Gubbay and Koppel claim that just like that paradox, the solution to many talmudic questions of probability will have more than one correct answer, depending on how you think about that answer.

Rabbi Nahum Eliezer Rabinovitch (b.1928) is the Rosh Yeshiva of the hesder Yeshivah Birkat Moshe in Ma'ale Adumim.  (He also has a PhD. in the Philosophy of Science from the University of Toronto, published in 1973 as Probability and Statistical Inference in Ancient and Medieval Jewish Literature.)  Rabbi Rabinovitch seems to have been the first to point out the relationship between Bertrand's paradox and talmudic probability theory in his 1970 Biometrika paper Combinations and Probability in Rabbinic Literature. There, the Rosh Yeshiva wrote that "the rabbis had some awareness of the different conceptions of probability as a measure of relative frequencies or a state of general ignorance."

James Franklin, in his book on the history of probability theory, notes that codes like the Talmud (and the Roman Digest that was developed under Justine c.533) "provide examples of how to evaluate evidence in cases of doubt and conflict.  By and large, they do so reasonably. But they are almost entirely devoid of discussion on the principles on which they are operating." But it is unfair to expect the Talmud to have developed a notion of probability theory as we have it today. That wasn't its interest or focus. Others have picked up this task, and have explained the statistics that is the foundation of  talmudic probability. For this, we have many to thank, including the Rosh Yeshiva, Rabbi Rabinovitch שליט׳א.

“(The [Roman] Digest and) the Talmud are huge storehouses of concepts, and to be required to have an even sketchy idea of them is a powerful stimulus to learning abstractions.”

— James Franklin. The Science of Conjecture: Evidence and Probability Before Pascal, 349.

[Repost from here.]

Zevachim 74a ~ Talmudic Probability Theory II

Our first deep dive into the math surrounding Talmudic Probability Theory was back in February 2015 when we were studying Ketuvot.  Today's Daf Yomi addresses the same theme, and thanks to some clever work by a contemporary mathematician we are able to provide some new insights today's page of Talmud.

Don't touch that goat

The reason we even get into this subject is a Mishnah that opens the eighth chapter of Zevachim, and which we studied just four days ago. There, the Mishnah rules on the problem of animals that must not be sacrificed getting mixed up with groups of those same animals that may be sacrificed. We have already learned that under usual circumstances, when a forbidden object falls into a majority of similar objects that are permitted, the prohibition is annulled, so that if one draws any object it is permitted (Zevahim 72a). However, the prohibition against idol worship is so strong that the usual rule allowing the minority object to be annulled in favor of the majority is not in play. This is made explicit in the definitive Shulchan Aruch, the Code of Jewish Law: 

שולחן ערוך יורה דעה 140:1

עֲבוֹדַת כּוֹכָבִים וּמְשַׁמְּשֶׁיהָ וְתִקְרָבְתָּהּ, אוֹסְרִים בְּכָל שֶׁהוּא, שֶׁאִם נִתְעָרֵב אֶחָד מֵהֶם, אֲפִלּוּ בְּאֶלֶף, כֻּלָּן אֲסוּרוֹת

Objects used for idol worship or their accessories are forbidden in any amount. So even if one of these objects was mixed into a group of one thousand, it is forbidden to derive any benefit from the entire group.

What about a 40:60 split?

Here is today's page of Talmud:

זבחים עד, א

אמר רב טבעת של עבודת כוכבים שנתערבה במאה טבעות ופרשו ארבעים למקום אחד וששים למקום אחר פרשה אחת מארבעים אינה אוסרת אחת מששים אוסרת

Rav says: With regard to a ring used in idol worship that was intermingled with one hundred permitted rings, and then forty of them became separated to one place, and the other sixty became separated to another place, so that they are now two distinct groups of rings, if one ring from the group of forty became separated from them and then became intermingled with other rings, it does not render them prohibited. But if one ring from the other sixty became separated from its group and became mixed with other rings, it renders them prohibited.

The talmudic reasoning here is not straightforward, (even with a lengthy attempt by Rashi) which is why we have to thank Rabbi Nahum Eliezer Rabinovitch (b.1928), the Rosh Yeshiva of the hesder Yeshivah Birkat Moshein Ma'ale Adumim.  (He also has a PhD. in the Philosophy of Science from the University of Toronto, published in 1973 as Probability and Statistical Inference in Ancient and Medieval Jewish Literature.)  In a paper published in 1969, Rabbi Rabinovitch reconstructs what he he calls "a plausible line of reasoning."

Rabinovitch N.L. Probability in the Talmud. Biometrika 1969, 56 (2): 437-441

As we have noted before, James Franklin, in his book on the history of probability theory, wrote that codes like the Talmud (and the Roman Digest that was developed under Justine c.533) "provide examples of how to evaluate evidence in cases of doubt and conflict.  By and large, they do so reasonably. But they are almost entirely devoid of discussion on the principles on which they are operating." But it is unfair to expect the Talmud to have developed a notion of probability theory as we have it today. That wasn't its interest or focus. Others have picked up this task, and have explained the statistics that is the foundation of  talmudic probability. For this, we have many to thank, including the Rosh Yeshiva, Rabbi Rabinovitch שליט׳א.

“(The [Roman] Digest and) the Talmud are huge storehouses of concepts, and to be required to have an even sketchy idea of them is a powerful stimulus to learning abstractions.”

— James Franklin. The Science of Conjecture: Evidence and Probability Before Pascal, 349.

[For more on Rabbi Rabinovitch and tamludic probability, see our post here, from where some of this is taken.]