I have no idea really. Judging by the number of people alive today (6.6 billion) we can estimate that the Earth is able to support perhaps around 1 billion large animals, so I would guess (and it is a pure guess) between 1-10 billion dinosaurs in total.
This illustrates an important point: there is no way of knowing the population of dinosaurs for certain (we cannot count them!), so we have to estimate it one way or another. Presumably palaeontologists and biologists have a model to estimate this, either based on the number of fossils found or the known parameters of the ecosystem at the time.
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danm commented on :
So if 1-10 billion dinosaurs were roughly alive millions of years ago, how many do you think were actually alive?