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."
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