Why would one doubt that Firefox has been downloaded 1 billion times?
The number itself doesn't tell us much. It only tells us that there have been 1 billion downloads of Firefox from the Mozilla web site in total -- representing all versions over several years.
It would take a particular type of dumb to a) interpret this as there being 1 billion instances of Firefox in use or b) to imply that anyone with an IQ of 80 or above seriously thinks this number represents the current installed base.
Only in relation to other metrics can this number be used to estimate anything, and even then you have to expect huge errors and that the model would have to change with what is happening in the ecosystem where Firefox exists. Playing with these numbers and coming up with a model only becomes science when it can predict and/or verify observations. Until then it is just an amusing fact.
As for the number itself; I am sure I have installed firefox perhaps 200 times (give or take 100, who's keeping track of these tedious things anyway?) over the past few years. New machines, machines that have been reinstalled, Firefox upgrades. Given a market share somewhere around the 20-25% mark it is entirely believable that over the years the Mozilla web site(s) have seen 1bn+ downloads.
So let's not read too much into this number. If you want to interpret the number then please have the courtesy of supporting it with a believable model and don't be a tedious fool.
It is an amusing fact and let's leave it at that.