From that video, it seems that Galton-Watson is the name under which such things are modelled. Thanks!
Jewish priesthood was hereditary, so they didn’t need to be individually “ordained”. Their status allowed them to perform Temple sacrifices and certain other rituals, but wasn’t thought to make them inherently wiser or better suited to give advice than anyone else. This priestly status still theoretically exists, and very Orthodox Jews who literally expect the Temple to return at some point feel it’s crucial to keep track of who is and isn’t a priest.
Rabbis, on the other hand, are an intellectual meritocracy. Towards the end of the Temple period a conflict arose between those who felt religious authority should lie with the hereditary priests (the Sadducees) and those who felt it should lie with the smartest and most pious guys around, regardless of their lineage (the Pharisees). Obviously, once the Temple was destroyed, the Sadducee position became untenable.
So they are entirely separate categories; a priest could also be a rabbi, but could equally well be a complete ignoramus. And no matter how learned a rabbi is, they will never acquire priestly status if they weren’t born with it.
Sure, they’re appointed by the Pope and Curia, but who actually consecrates them? Not necessarily the Pope.
Take Daniel Cardinal DiNardo of Houston- he was consecrated by Bishop Soens, with Cardinals Wuerl and Burke assisting.
This Brittanica article indicates (emphasis mine):
Cardinals in the Catholic Church are nearly always also bishops.
Thanks for that.
From this article
A corollary of high extinction probabilities is that if a lineage has survived, it is likely to have experienced, purely by chance, an unusually high growth rate in its early generations at least when compared to the rest of the population.
Which seems to apply to the case of the bishops.
I did not say explicitly before, but (as a crude model for some of this stuff) one can consider a Wright–Fisher diffusion model, or some generalization. But this requires some thought about what is the correct model and parameters if we are going to infer anything about real historical data.
I think, however, that the involvement of the Pope is limited to the appointment of the candidate to the office of a particular diocese. The sacrament of ordination that makes the candidate a member of the class of bishops, with all the sacramental powers of a bishop, is done by another bishop, not by the Pope (at least not in his capacity as Pope - of course the Pope is himself a bishop and hence able to conduct ordinations just like any other bishop).
Also note that it is not uncommon for someone who is already a bishop to move to another diocese (or even to his first diocese, if he was previously a titular bishop who did not actually run a diocese). In that case, the papal appointment to the new diocese takes place, but there is no new ordination.
Cardinal, Archbishop, etc… aren’t higher levels of ordination, they’re more like administrative ranks. There are only 3 levels of ordination/consecration - deacon, priest and bishop. The Pope is the Bishop of Rome, which is basically first among equals, and by virtue of the office, the head of the Church. He can consecrate bishops and ordain priests and deacons, and so can any other bishop.
The actual selection of the bishops to the sees is part of the Church administrative organization (the Curia) and the Pope is over that. But the actual religious part of it is usually done by local bishops.
It’s kind of like how a medieval knight or lord could knight anyone they liked, but whether or not that new knight had titles, lands, etc… was up to the Crown if they didn’t already stand to inherit.
In fact, an Israeli I know once told me that without the Temple, there can be no priests, but without priests, no Temple can be consecrated.
Getting back to the OP, the mathematical principle is clear. That is why the “Eve hypothesis” that there is some female hominin from whom we all descended is clear. (There is a corresponding “Adam” hypothesis. But absolutely no reason to think they ever met or were even contemporaneous.)