In an astrological chart, there are twelve possible signs or constellations. There are also seven major heavenly bodies (Sun, Moon, Mars, Mercury, Jupiter, Venus, Saturn), plus the ascending and descending lunar nodes, each of which may be in one of twelve houses.
This makes a total of ten degrees of freedom, each of which may take on twelve possible states. There are therefore 12^10 ≅ 62 billion possible charts.
Suppose that, first, reincarnation exists; and second, charts persist across lifetimes. After a person dies, their soul is shelved until the next birth with a matching chart becomes available, whereupon they are automatically resleeved.
How many people alive today are on their first life?
The current population of the world is approximately 7 billion. The number of people who have ever lived is not known precisely, but is estimated at about 107 billion, implying about 100 billion dead.
Since more people have died than there are charts, the number of reincarnations must be greater than zero. (More precisely, the lower bound is somewhere around 38 billion.)
It is possible to have more souls than charts, since identical twins may have the same chart, but require separate souls because their lifespans occur in parallel rather than in series. However, this doesn't move our lower bound by very much, because only about 1-3% of successful human pregnancies result in multiple births. So the true lower bound might be around 37 billion.
We don't want lower bounds, though; we want expected values. How to calculate this?
Consider a given newborn. They have a certain chart; what are the odds that no one has ever had that same chart before? There are 100 billion souls-in-waiting, and each has a 1 in 62 billion chance of hitting that chart exactly. So the odds that they all missed are (1 - 1/62 billion)^100 billion ≅ 20%.
Naively, this implies that about 20% of people being born today are new souls, and about 80% are reincarnations. However, if 2% of successful pregnancies result in multiple births, then about 1% of births are second or later in a set of multiples. (Not all twins-or-more will share the exact same charts, but if they're born within ~15-30 minutes of each other then they probably usually share.)
Triplets and more are sufficiently rare to round to zero. Second twins, then, are about 1% of the population, and therefore have 1% as many chances to find a predecessor – since a second twin can only be a reincarnation if there was previously a different pair of twins that shared the same chart. So a second twin's chance of newsouling is (1 - 1/62 billion)^1 billion ≅ 98.4%. For our purposes, we can basically round this off to 100%.
1% of the population is newsouled due to being second twins, and 20% of the rest of the population is newsouled by chance. This implies about 21% of the population are on their first life.
What about unrelated births that happen to occur in close succession? The global birth rate is about 16,000 people per hour; surely charts can't change that fast?
The houses of an astrological chart are apparent directions in the sky, relative to the ground. For example, the first house is on the eastern horizon, and the fourth house is directly overhead. When the sun is in the seventh house, it's sunset.
In the short term, then, the movement of the things through the houses is dominated by the rotation of the Earth, and so each variable changes about every two hours. Multiplying by ten variables, that's an average of about 12 minutes between changes.
In addition, even simultaneous births can have different charts if they occur in different time zones. We can divide the Earth radially into twelve astrological time zones, according to which house the Sun appears in from that location; assuming that matching charts must be both within 12 minutes and in the same twelfth-slice geographically, this implies that we can get an average of one unique birthchart per minute.
The global birth rate is 265 people per minute.
This completely shatters all of our previous calculations.
Taken naively, it implies that something like 99.6% of the population is newsouled. This isn't actually reasonable, though, because this sort of high birth rate has been going on for a while, so the tenth or fiftieth person born to a given chart could easily find a soul waiting for them.
The core of the problem is that birth charts are not independent trials. How long does it take for a chart to recur?
Probably about 12^10 minutes, or 118,000 years.
Estimating the prehistoric world population is difficult, but the available data make me fairly confident that the population a hundred thousand years ago was well under ten million: for our purposes, approximately zero.
Final answer: we have not yet reached the age of reincarnation. The only way anyone is getting reincarnated nowadays is if an infant dies within a couple of minutes after being born, so that it can be born a second time within the same ~12-minute window.
Either that, or there are several thousand reincarnated Neanderthals and Homo erectus hidden among us.