Suppose you were transported back to the late Cretaceous, what is the probability that you would see a Tyrannosaurus? That depends on a large number of factors, and to simplify, I shall limit myself to T Rex. There were various Tyrannosaurs, but probably in different times and different places. As far as we know, T Rex was limited to what was effectively an island land mass known as Laramidia that has now survived as part of Western North America. In a recent edition of Science, a calculation was made, and it starts with the premise, known as “Damuth’s Law” that population density is negatively correlated with body mass through a power law that involves two assignable constants, plus the body mass. What does that mean? It is an empirical relationship that says the bigger the animal, the fewer will be found in a given area. The reason is obvious: the bigger the animal, the more it will eat, and a given area has only so much food. Apparently one of the empirical constants has been assigned a value of 0.75, more or less, so now we are down to one assignable constant.
If we concentrate on the food requirement, then it depends on what it eats, and what it does with it. To explain the last point, carnivores kill prey, so there has to be enough prey there to supply the food, AND to be able to reproduce. There has to be a stable population of prey, otherwise the food runs out and everyone dies. The bigger the animal, the more food it needs to generate body mass and to provide the energy to move, however mammals have a further requirement over animals like snakes: they burn food to provide body heat, so mammals need more food per unit mass. It also depends on how specialized the food is. Thus pandas, specializing on eating bamboo, depend on bamboo growth rates (which happens to be fast) and on something else not destroying the bamboo. For Tyrannosaurs, they presumably would concentrate on eating large animals. Anything that was a few centimeters high would probably be safe, apart from being accidentally stood on, because the Tyrannosaur could not get its head down low enough and keep it there long enough to catch it. The smaller raptors were also probably safe because they could run faster. So now the problem is, how many large animals, and was there a restriction? My guess is it would take on any large herbivore. In terms of the probability of meeting one, it also depends on how they hunt. If they hunted in packs, which is sometimes postulated, you are less likely to meet them, but you are in more trouble if you do.
That now gets back to how many large herbivores would be in a given area, and that in turn depends on the amount of vegetation, and its food value. We have to make guesses about that. We also have to decide whether the Tyrannosaur generated its own heat. We cannot tell exactly, but the evidence does seem to support the fact that it was concerned about heat as it probably had feathers. The article assumed that the dinosaur was about half-way between mammals and large lizards as far as heat generation goes. Provided the temperatures were warm, something as large as a Tyrannosaur would probably be able to retain much of its own heat as surface area is a smaller fraction of volume than for small animals.The next problem is assigning body mass, which is reasonably straightforward for a given skeleton, but each animal starts out as an egg. How many juvenile ones were there? This is important because juvenile ones will have different food requirements; they eat smaller herbivores. The authors took a distribution that is somewhat similar to that for tigers. If so, an area the size of California could support 3,800 T. Rex. We now need the area over which they roamed, and with a considerable possible error range and limiting ourselves to land that is above sea level now, they settled on 2.3 + 0.88 million square kilometers, which, at any one time would support about 20,000 individuals. If we take a mid-estimate of how long they roamed, which is 2.4 million years, we get, with a very large error range, that the total number of T. Rex that ever lived was about 2.5 billion individuals. Currently, there are 32 individual fossils (essentially all are partial), which shows how difficult fossilization really is. Part of this, of course, arises because fossilization is dependent on appropriate geology and conditions. So there we are: more useless information, almost certainly erroneous, but fun to speculate on.