While it's true space is quite empty, I'm sure before sending it on it's way NASA did some calculations to make sure it wasn't going to cross the path of any known interlopers (for lack of a better term). I doubt they would just send it off in any random direction and hope for the best.
Question: When we send spacecraft through the asteroid belt to the outer planets, how do we navigate the craft through the belt?
Answer: Pioneers 10 and 11 had preceded the Voyagers to Jupiter and the asteroid belt was a major concern for them. By the 1960's more than 3000 minor planets had been discovered and their orbits well determined. Even 50,000 minor bodies spread over the volume of space occupied by the asteroid belt would produce little direct danger, although a chance collision with an uncatalogued object was possible.
"While the largest of the asteroids were known and their orbits charted, many of the asteroids moved in unknown orbits. Although the risk of a spacecraft colliding with a charted asteroid was negligible, there was no way to estimate how many particles the size of a grain of sand might be present in the asteroid belt to collide with the spacecraft and seriously damage it". (From Pioneer, First to Jupiter, Saturn and Beyond, NASA SP-446, 1980) Only by going there could the danger be properly assessed - and Pioneer was first.
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Even something the size of a grain of sand could potentially damage a craft traveling at the speeds required for space exploration and Voyager has gone past Pluto without encountering any. Space is really, really empty. Even if you did zero calculations whatsoever, your chances of crashing into something are virtually non-existent. But you're right, we probably wouldn't send up anything without doing some pretty extensive calculations beforehand, all of which run counter to the OP's point that we're going to crash into something in space.