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