Tuesday, February 19, 2013

Brit milah math

The Mishnah on Shabbat 137a discusses a mohel who mistakenly circumcised a baby on Shabbat when the baby was either seven or nine days old:

משנה. מי שהיו לו שני תינוקות, אחד למול אחר השבת ואחד למול בשבת. ושכח, ומל את של אחר השבת בשבת - חייב. אחד למול בערב שבת, ואחד למול בשבת, ושכח ומל את של ערב שבת בשבת, רבי אליעזר מחייב חטאת, ורבי יהושע פוטר.

The Gemara then presents different versions of this mahloket tanna'im, changing which cases are agreed as hayyav, agreed as patur, or disagreed.

According to those who say that circumcising a baby on his seventh day makes you hayyav but on his ninth day makes you patur, it sounds that there is no mitzvah of milah at all before the eighth day. I might have thought that there is a mitzvah to circumcise from birth, and the eighth day is just the proper zeman.

Indeed, Yalkut Bi'urim cites aharonim (Shakh, YD 262:2; Mishneh la-Melekh, Hilkhot Melakhim 10:7; Panim Yafot, Lev. 5:2) who claim based on rishonim that a baby circumcised before his eighth day requires hattafat dam berit once he is eight days old.

The Beit ha-Levi famously separated the mitsvah of the act of circumcision from the mitsvah of the state of being circumcised. One of the authors of the Yalkut Bi'urim suggests that having the baby circumcised on the seventh day fulfills the latter mitsvah, but not the former, explaining why he would be patur on Shabbat. The hiddush there I suppose is that the mohel also fulfills the latter mitsvah himself, and not just the baby, since the mohel is the one we are worrying about being hayyav or patur.

This amud is incredibly rich for iyyun. How does the mitsvah of milah change between days seven and eight? How does this change relate to dehiyat Shabbat? What factors do we consider for the hiyyuv/petur regarding Shabbat after the fact?

Now, I promised you some math.

Here's how this gemara ends:

תני רבי חייא, אומר היה רבי מאיר: לא נחלקו רבי אלעזר ורבי יהושע על מי שהיו לו שני תינוקות אחד למול בערב שבת ואחד למול בשבת, ושכח ומל את של ערב שבת בשבת - שהוא חייב. על מה נחלקו - על מי שהיו לו שני תינוקות, אחד למול אחר השבת ואחד למול בשבת. ושכח ומל של אחר השבת בשבת, שרבי אליעזר מחייב חטאת ורבי יהושע פוטר. השתא, רבי יהושע סיפא דלא קא עביד מצוה - פוטר, רישא דקא עביד מצוה מחייב? - אמרי דבי רבי ינאי: רישא - כגון שקדם ומל של שבת בערב שבת, דלא ניתנה שבת לידחות. סיפא - ניתנה שבת לידחות. אמר ליה רב אשי לרב כהנא: רישא נמי, ניתנה שבת לידחות לגבי תינוקות דעלמא! - להאי גברא מיהא לא איתיהיב.

Rabbi Hiyya had a version of Rabbi Meir where circumcising a 9-day-old baby on Shabbat is worse than circumcising a 7-day-old baby on Shabbat. The students of Rabbi Yanai explained that in the latter case, at least there is a baby present who is actaully eight days old, whereas in the former case no baby present requires circumcision. (Apparently the Shakh et al. would have to say that hattafat dam berit in such a case is not doheh Shabbat, or at least doesn't qualify as a makom mitsvah, at least for Rabbi Meir. Also, Rabbi Yanai's students' explanation fits well with the fact that there are always two babies in the case instead of just one.) Rav Ashi asks, but why not say that the latter case should also be patur, because there are other babies out in the world that need to be circumcised?

This raises a question: how likely is it that a Jewish baby somewhere out there is having his bris this Shabbat? What does the birthrate need to be for the odds to be, say, 99 percent?

Fortunately, the hard work was already done in 1837 by Siméon Denis Poisson. A Poisson distribution describes the chance of a random event X happening x times within a certain timespan if we know the long-term average μ (Greek letter mu) for that timespan. The equation happens to be P(X = x) = exp(−μ) μ x / x!.

If we want the odds of at least one bris on Shabbat, that's the equivalent of the probability of not having zero brisin on Shabbat, or P(X > 0) = 1 − P(X = 0) = 1 − exp(−μ). We'll call the average number of brisin per day μ, but since brisin per year is easier to think about, we'll substitute μ = N / 365, where N is brisin per year. So all together, our equation is p = P(X > 0) = 1 − exp(−N / 365).

Here's a plot of that equation, p vs. N:

The x-axis is average brisin per year, and the y-axis is the probability of a bris this Shabbat, as a fraction of 1. The dotted lines there mark 50, 90, and 99 percent probability.

The odds of a bris in the world this Shabbat are 50-50 if an average of 253.00 healthy male Jewish babies are born every year. The odds are 90 percent if the average is 840.44 babies per year and 99 percent if the odds are 1680.89 babies per year.

That's for an individual Shabbat. What if we want to know the odds for a bris happening on every single Shabbat of the year? If that means 52 Shabbatot in a row, just raise the entire expression to the power of 52, or p = P(Xk > 0) = (1 − exp(−N / 365)) ^k for an arbitrary number of Shabbatot k in a row.

The plot, p vs. N, with k = 52:

The odds are 50 percent if 1578.41 healthy male Jewish babies are born per year, 90 percent for 2236.96 babies per year and 99 percent for 3121.29 babies per year.

If we want the hazakah to be really strong, let's demand a bris somewhere in the world every Shabbat for 100 years straight. That's 5200 Shabbatot in a row, let's say. We'd need a much higher birthrate, right?

Not that much higher, since logarithms are involved: N for a given probability p is N = −365 log(1 − p ^(1 / k)), for k Shabbatot in a row, by solving the above equation of p for N. To keep the odds at 99 percent, you just need to raise the male baby birthrate to 4802.15.

The plot, k = 5200:

Here's the probability of keeping up the streak for the whole millennium, k = 52000:

As a final bonus, here's a log-linear plot that extends the brit milah streak almost 20 billion years into the future. The x-axis is number of Shabbatot in a row, k; the y-axis is the required birthrate, N; and each plot line represents a given probability, p:

Some back-of-the-envelope calculations based on data from this report puts the current worldwide Jewish boy birthrate in the ballpark of 100,000 births per year. I think it's safe to say mazal tov! And mazal tov for the baby girls too!

Finally, a bit of trivia, which might be confusing at first glance. We commonly circumcise our baby boys when they are just 6-point-something days old. How is that? We circumcise a baby on his eighth day of life, meaning that a baby born on Sunday is circumcised the following Sunday. If the baby was born in the afternoon, and the bris is in the morning, then the baby saw only a fraction of two of his eight days, so his total lifespan at the time of the bris can be less than seven days.

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