Average of what population?
I am sure ~64 years is generous for the population as a whole, but if you have 50% infant mortality, that average is not a very useful figure.
I can believe in ~64 years life expectancy at puberty for instance. And that is a much more useful figure to understand demographics of the game world.
based on the dice rolls worst case is someone rolls an exploding die at age 35, so death by old age at 35 is possible, but presumably rare.
If you calculate from a roll of 5 every year they will start getting aging points at age 50. ignoring explosions between 30 and 40 (5 years, roll over 6 results in aging) they will on average get 2 aging points during this time, and 5 aging points between 40 and 50. On the other hand every year there is a 1 in 10 chance of crisis, and reaching the next level of decrepitude as well, so the easier calculation is one crisis per decade, putting an affective average age at 85 from simple gaining of decrepitude to level 5. realistically unless they are allowed to rest after reaching decrepitude 4 this will be 75, if they survive the crisis. If we assume te first crisis at 45 (roll+5) then they only need a survival roll 1 time in 10 and will have to make a stamina roll of 3+, so minimal losses. At age 55 they will have their second crisis and have to roll at +7, with a 30% chance of having to make a roll for survival, so probably about 15% of the population would die at this age. at 65 the third crisis will roll at +9, meaning a 50% chance for a survival roll, though of those making that roll I would expect about 10% to survive, so 45% loss of the remaining 85%, or about 38% losses for 53% dead at this point, so 65 would at least be the rough median age.
I stated several times that I tried not to include infants and babies (by which I mean children under 5) in the calculations. It greatly complicates the math for little gain. Since we know the rough population growth rate, we can calculate how many have to live to adulthood no matter what the [birth rate - youth mortality rate] actually is.
Infant mortality was sub 30%. Youth mortality (meaning all deaths below 15) was actually at around 50% for most of human history until you get to the late 1800s/early 1900s where it began to decline.
It is just easier to exclude the infant/youth deaths and lower the total births to equal the growth rate. Including them also screws up numbers like the average life span for your population if you want to know how long your actual Covenfolk will live to.
The ~64 I gave was done with math similar to Silveroak's. I guess we could call it the "Adult Life Span". If you want the actual "Life Span" for the entire population including infants, you have to reduce the number by roughly the infant death rate, the add ~7 [the mid point of Youths] and reduce that by (youth death rate - infant death rate). Those rates during 1220 are high 20s (27~28%) for infant and high 40s (~48%) for youths. From there with adults living to ~64, we get an actual "Life Span" of ~42.5 years. If you want the number most commonly seen as "Life Span" you would just add ~7 to the total (so ~64 + ~7 = ~71) and reduce it by youth mortality rate. This gives a fairly historically accurate lifespan for the middle ages of just under 37.
Doing the math in reverse from the actual middle age life span and youth death rate gives an average adult lifespan of just under 61. However there are lots of small fluctuations and ~64 was the number that my math with the AM system produced. Since they are so close, using the number based on the system rather than reality (since we are talking about a game) just felt right.
And yes I did take into account that the actual birth rate is double in the few places it mattered. But in general it didn't, since we have a rough percentage of the population for each age group.
Man, this feels really dark...
Sorry I totally missed your post and it actually is something that I would be more inclined to use. I know both of us disagree with the 1:10k ratio that has been in place for most of AM's history from a prior conversation.
If the estimated number was wrong and it was actually something like 1:5k then that would make things far easier. Rather than being right at borderline to meet the apprentice requirement with some (not insignificant) portion of Redcaps actually searching for Gifted children, there would be enough that it might only require one or two Redcaps per large Tribunal tasked with the duty and the rest are found by individual Magi.
I wrote an article called "Demography of the Order of Hermes" in SubRosa #14, where I applied my professional experience as a population modeller to this very issue. I gathered data from the HoH books and estimates of mortality rate to make predictions using a Lefkovich matrix model. There is a section on the article entitled "Supply of Apprentices" where it reports that to support its current rate of growth, the modelled order requires 186 Gifted children over the next seven years.
There might still be copies of SR #14 hanging around somewhere if you'd like to know what I did; I've probably got my original manuscript version on a computer somewhere...
Mark
Do you remember your estimate of the current population of the order? I've been looking at the section in the hibernia book that states that 7 gifted children are presented at each tribunal and thinking that looks rather high, but I'm guessing from your modelling that it's rather the opposite and it might be a bit low?
I have a table that gives the current size of the order, based on the three HoH book, each of which says the size of their house. The total comes to 1240 magi.(approximately, since exact numbers of Ex Misc and Mercere are not given).
My model started with the 13 Founders in 767 AD and with some basic assumptions about 'births', 'deaths' and immigration, predicted 1267 magi in 1222 AD.
Note that Hibernia has one of the lowest populations amongst the tribunals, at only 0.6 million (compared to 10.4 million in Rhine -- data also presented in SR #14), so I wouldn't have any issue with Hibernia having half the apprentices than the average. I should also note that I had written the model presented in the SR long before the writing of The Contested Isle, and we'd been using its output for tribunal books for several years. 
Mark
Note as well that at the standard 1 gifted per 100,000 Hibernia, with a total population of 600,000 should have 6 Gifted individuals total Being able to contribute 7 apprentices per tribunal (1 per year) suggests a significantly higher percentage of the population. Within the average lifespan of 65 years there would have been 65 Gifted individuals given to the order, and presumably more than that which are either found by mages or kept for hedge traditions which apparently rival the Order in population. So if the population of 600,000 has a Gifted population of150 that is still 1 in 2,000 which is far above the established standard.
So the Order needs to find about ~26.5 Gifted in the age range appropriate to become children every year. That is, if they are going to maintain their current rate of growth. Don't know why I thought it was 10 per year.
Which even with the minor magic support lowering the difficulty by 3 is not nearly enough. Even if that 30 devoted both seasons a year (rather than one searching and one delivering letters) they would not reach 60% of the required number. And we actually need to find more than the ~26.5 per year, since not all found will be appropriate to become Magi (poor Int, bad Flaws, etc).
So I broke down the math and the exact percentage for Per 0 or +1 (it would have to be +2 to change) is 5.9% per season at EF 12+. If the EF drops to 9+, the success rate raises to 21.9% per season (actually somewhere between 16.9% and 28% based on how many have Per 0 and +1).
I do not believe we can justify reducing the EF lower, since the 9+ would come from a combination of both Magical Device assistance (the smaller contribution to the lower EF since Magi will be using spells for this if they search) and the fact that the Redcaps are not fully vetting the apprentices but only noting that they have the Gift (the larger contribution since Magi would be carefully checking out any they find to determine if they would be a suitable apprentice).
If we add in 26.5 Magi searching per year (to meet the required amount needed) it will not really move our total found up much (~1.6 per season) since the Magi will be searching at the higher EF for Gifted appropriate for their apprentice. Also they would have to spend some time going through the potentials found by the Redcaps.
Looking at it from the other end, if Magi are doing the search totally by RAW, we would need a total of ~449 seasons of search per year. That would be over a third of the entire Order spending a season searching each year or over 112 Magi spending all four Seasons searching. If we set the average Per in the Order to +1, you are looking at a Magi having to search for 17 Seasons to find an apprentice. This is completely unsustainable the way AM is currently played, which is why I started this thread in the first place.
We need a total of ~94.6 seasons of search per year at the lower EF 9+ (with Magi Per average at +1) to find enough Gifted. Any Magi who wanted an apprentice would have to search for a year to find one. This is assuming any found apprentice is acceptable (not low Int or with disqualifying Flaws, appropriate age range, etc). If you want to ignore everything else in this thread, the Find Apprentice EF should be reduced to 9+. This is actually a change that should be considered as Errata.
If we are using the suggested 9+ EF for Magi doing the search to find an appropriate and acceptable apprentice, we can use a lower 6+ EF for Redcaps who are just finding Gifted children and cataloging them. To find one appropriate as an apprentice, Magi would still have to spend a season or two checking them out (travel time, interviewing, etc).
At this rate, Redcaps with appropriate magical items would have between a 49% (Per 0) and 59% (Per +1) chance per season. We would need roughly 50 rolls per year. There are different ways you could break this down.
How many they would need to find would be based on what percentage found would be appropriate to become an apprentice. If we figure 2/3s, they would have to find ~40 a year. If we figure 1/2, they would have to find ~53 per year. If we figure 1/3, they would have to find ~80 a year.
A lowered base EF of 9+ for everyone would be the easiest fix if you do not want to change anything in the current play of the game. If you do not like the idea of Redcaps searching out and cataloging Gifted in your Saga, it is pretty much a required change to allow the Order to continue on the path it has by RAW. @David_Chart you might want to consider this.
Sounds good to me!
Seriously, making the demographics work for an RPG is really hard. The worst case I am aware of is Blue Rose, where, based on the fiction, only one-third of the adults in (nice, gender-equal, LGBTIQ+ inclusive) Aldis are in potentially fertile relationships, and two or three children per fertile relationship seems to be normal. Demographic collapse should have taken them out centuries ago. I can't think of any way to fix that problem that would not also basically destroy the artistic goals of that part of the setting, either. So handwaving demographics for gaming is good.
Let me check that I have the reasoning right.
We know how many apprentices the Order needs. Given an EF of 12, they cannot find that many, no matter what we say about the background demographics. Therefore, we should drop the EF to 9, for background consistency.
That also sounds good to me. I'll do it.
I can't argue with the math, but a roll of 6+ seems a little too easy. Especially if the Redcaps in questions are (self-) selected for being perceptive. Given that the Mercere run a breeding program, getting a couple of lines of Per +3 apprentice hunters seems entirely reasonable (buying Per +2 and then getting another +1on top from a minor virtue something-blooded).
I do think that it makes a lot of sense that some redcaps would be doing this as a job, since a good apprentice would fetch a decent amount of Vis. I also suspect that it would skew more towards the more organised Tribunals (Thebes in particular) and towards gifted people living in cities or along the main travelling paths, as there are higher population densities and more passage.
A knock on effect of this would be to likely distort the background of the mages in the order: a Redcap can run around populous France, the HRE or in the North African trading centres and see many humans to scan. The same amount of time in low density Finland or Sweden would not be as efficient so likely ignored. Which would distort the background of the Order's population in favour large "nations" (kind of anachronistic term, but it was used by the University of Paris in the middle ages).
This is what I dislike of there being so many Tribunals in the British isles: the small mundane population compared to the hermetic population. So the only way for the Order to maintain itself there needs to be fueled by importing Apprentices from the more (densely) populated area or by Magi moving there post-gauntlet because their Tribunals are over crowded.
Maybe. It would also be plausible if the Gift is more widespread, occurring in a larger proportion of the population, in those remote areas where the dominion is weaker and the magic realm is stronger.
(Now, one could of course wonder if this is the case for Hibernia, that the dominion is relatively weak. The Church has a long history there, and she is not particularly sparsely populated. However, she is canonically extremely rich in vis, and I imagine the density of the Gift would be similar. Possibly a canonical anachronism. I don't know.)
Back to the question at hand, I very much agree with @David_Chart . Ars Magica is not a numbers game, and the numbers do not have to work to make the narrative plausible. The ratio of the Gift in the population does not show up in the narrative (except possibly for a very atypical sort of sagas). The story will never tell you if it is 10^-6 or 10^-3, especially if some go unnoticed. What is important to the story is how hard it is to find one. It should not be impossible, or likely to take years, nor should it be an automatic downtime success. It might be for some parens, but those are not the ones we tell stories about. That is much more important when setting ease factors than a demographic analysis.
Something as important to the story as getting an apprentice, or say having a child, the SG should decide "Do I want this or not?" and make it happen, ignoring the dice.
As there is a number in a book somewhere I guess Atlas should try to be as accurate as possible. Any future publications, I recommend not writing any number which will let you be pinned down on demographics like this. 
Second that. We generally add an apprentice when a player wants to play one ...
Second the other half as well btw.
I agree with both of these, the most important is to get a fun/interesting story out of it.
I don't think the EF needs to be changed. As stated by others, an apprentice is easy to introduce through a story.
I always saw the EF as "if you don't want a story around this, pay someone to look for an apprentice and he should get you a suitable child in no more than 16 seasons, give or take".
Also, statistic models are a lot better at explaining how things happened in the past then at predicting how they are going to be at the future. As I understand, Mark's model shouldn't be read as "the Order will find 186 apprentices over the next seven years", but instead, in exactly the words he has used, which I quote:
to support its current rate of growth, the modelled order requires 186 Gifted children over the next seven years
If EF +12 doesn't fit with this, I see it as a feature, not as a bug. The predicted outcome for the Order is that if it is relying solely on people looking for apprentices it will dwindle and fade.
The exact development of the Order based on the assumed numbers is better left for each troupe to decide.
What we should look at is, EF +12 to look for an apprentice means that it takes on average about 16 seasons for someone with perception 0 or 1 to find a suitable apprentice. From a purely mechanical standpoint, does this improve or detracts from the game?
EF +9 means this would drop to at maximum 6 seasons for Per 0, and less for very perceptive searchers (about 3 seasons for Per +3, no more than 1 or 2 for Per +4 or +5). Do this supports a better gaming experience? It's an honest question, I don't know the answer... But...
I like that with the current rules the magus must put some effort in finding an apprentice. He can either accept a story, or hire someone, or divert resources to look for an apprentice and wait a few years (for example, the Jerbiton asks his merchant contact to keep an eye open for strange kids. He will bring someone to the magus after looking for 16 seasons, but he can't search and keep surveilling the baron at the same time).
If the result of changing the numbers makes finding an apprentice so much easier, why not just say "if you want an apprentice just say so and your SG will give you one without any effort at all, don't sweat about it"?
On the other hand, why not keep the numbers as they are and assume that there are about 1 person for every 5 magi (almost one per covenant) working for the Order (not the Order as an institution, but instead, working for magi of the Order) looking for apprentices through Europe?
I think it becomes more of an interesting question when you factor it by houses- if Bjorner has a harder time recruiting because they stick to the low population wilderness while Jerbiton blossoms searching in cities where they compensate for the dominion auras, and some traditions of ex Miscelania fade away even as the house hangs on through sheer bulk numbers then you can dig up all kinds of stories based on those numbers.