This weekend I first heard that according to Jewish tradition, the obligation to “be fruitful and multiply” requires one to have a minimum of one girl **and **one boy. This claim seems to check out. This got me thinking: What is the effect of adhering to this norm on the expected number of Jewish children?

If the norm just said “Have two children,” then the expected number of children required to satisfy it is of course two. But the norm says “at least one girl and at least one boy.” So assuming the gender of each kid is always a coin toss, you have a 50% chance of meeting the obligation with 2 kids (BG, GB). If you fail, you have a 50% chance of meeting the obligation with your third kid (BBG, GGB). If you fail again, you have a 50% chance with your fourth kid (BBBG, GGGB).

Looks like we have an infinite sum for the expected number of kids required to meet the Jewish norm:

.5*2+.25*3+.125*4+…, which converges to 3.

**Proof:**

Let x=.5*2+.25*3+.125*4+…

Then .5x=.25*2+.125*3+…

x-.5x=.5*2+.25+.125+…

.5x=1.5

x=3.

Thus, the effect of this norm relative to the “any 2 kids” norm is to increase the expected number of children by 50%.

I’m on the record in favor of having more kids. I believe that, in most cases, both individuals and society would be better off if families had three or four. A lot of people have small families because they are mildly tired when they are young, and fail to consider that as a result they will be very lonely when they’re old. Two grown children is not enough to get a decent quantity of phone calls and grandchildren.

So it’s hard for me not to sympathize with a rule that raises the expected number of children to three. However, the rule also raises the *variance *of the number of children. If people took it seriously, almost 1% of families would have seven kids! Even I doubt that’s a prudent course of action. The rabbis would have been wiser to prescribe three or four children total, regardless of gender.

Wiser still, though, would be “Naively weigh the costs and benefits of children to figure out the number that would make you happiest. Then double it.”

## READER COMMENTS

## Matt McIntosh

## Jul 31 2005 at 9:50pm

Phew, I don’t have to change my mind then according to your rule: when you double zero, it’s still zero. 🙂

## Phil

## Aug 1 2005 at 8:55am

Easier proof that the expected number is 3:

Have one child. You then need a child of the opposite sex. Since the probability is 1/2, the expected number of tries to get the correct second child is 2.

1 try for the first child plus two tries for the opposite sex child equals 3.

## Robert

## Aug 1 2005 at 9:38am

Of course, it should be borne in mind that this is the expectation of the number of children to minimally meet the rabbinical interpretation, and that there is no reason to expect that the 50% of couples who meet the requirement after two children will stop there.

## Neal Phenes

## Aug 1 2005 at 12:35pm

I have a boy and a girl. It is called a “rich man’s family”. I accepted the term but was unsure what was meant by that. I now think that, since I can “be fruitful and multiply” after having the minimum of 2 children, I will have more money available after expenses than I would have if I had to pursue and raise more and more kids.

## Robert Schwartz

## Aug 1 2005 at 7:01pm

“The rabbis would have been wiser to prescribe three or four children total, regardless of gender.”

When our sages, of blessed memory, were prescibing, having two children survive childhood, was a much chancier business than it is now.

Our own heuristic is one for me, one for you, and one for the Holocaust.

## jaimito

## Aug 2 2005 at 2:42am

One of each kind is a soft, Reform interpretation of the Law. “One for the Holocaust” should mean: one boy + one girl + two for assimilation + four for the Holocaust. Even so we shall never reach our pre-Holocaust numbers.

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