The evil lord Darth Vader, obsessed with finding young Skywalker, has dispatched thousands of remote probes into the far reaches of space….
This isn’t just unlikely to work, it’s so unlikely to work that it’s not even worth considering. Have you ever tried to find one person out of one hundred quadrillion? If so, did you even think about searching by brute force? “Nope, that’s not him. Next?” Those droids simply aren’t capable of finding Luke just from being on the same planet. Darth Vader could probably do it, thanks to magic, but those droids can’t.
Hoth has a surface area of about 160 million square kilometers, all of which is ice. That’s more space to search than the land area of Earth. The droid is probably smart enough to aim for the equator, since if people are on that planet they’re more likely to be where it’s only mind-bogglingly cold. Even so, it’ll have to search through an awful lot of nothing to find the Rebels. From what we see, it’s not equipped to travel across the world, and it didn’t know where the Rebel base was until after it landed. This isn’t Phineas and Ferb, so naturally, the movie-makers handwaved the problem by having the droid land within visual range of the very person it was looking for. Starting off with a coincidence of that magnitude is not a good sign. But that’s not even the biggest problem.
A galaxy has an awful lot of planets. According to here, there are probably billions of habitable planets in our galaxy. (And he shows that using simple math that even Star Wars writers could understand.) According to Wookiepedia, that galaxy has an awful lot of planets, of which about a billion are inhabited. That means that the odds of a probe droid even hitting the right planet are a literal one-in-a-billion shot.
But it’s even worse than that! They aimed probe droids at the uninhabited planets, too. We know that because Hoth was, as far as they knew, uninhabited. So they had to find one guy out of not one planet, not two planets, but one hundred and eighty billion planets. Obviously they got really lucky when the droid landed on a random point on the planet and just happened to be within visual range of the very person they were looking for, but that’s not even the biggest problem. The much bigger problem is that even if the droids could instantaneously search any planet they landed on, they’d have to get there first.
We could give the Empire the benefit of the doubt. After all, they’re a civilization capable of blowing the Olympian gods out of the water. Maybe they used enough droids to have a chance? Well, let’s check the numbers. Episode V takes place three years after the Battle of Yavin, so Vader has been looking for Skywalker for no more than three years. To land a droid on every planet, they’d need sixty billion droids per year. Maybe they can do that? I have no idea what the Empire’s manufacturing capabilities are like. Let’s find out.
A probe droid is 1.6 meters long, so one hundred and eighty billion of them in one place would cover 1.6^2*180*10^9=461 billion square meters. We’re talking enough area to cover Austria. Five times.
Put another way, if that number of probe droids were all flying in formation, it would take up a volume of 1.6^3*180^9=737 billion cubic meters. If they were arranged in a sphere for some reason, that sphere would be about 5.6 kilometers in diameter. That’s a lot of droid. Remember the asteroid Apophis? The one that had an unusually high chance of hitting Earth and got way more publicity than it deserved? It was .32 km across, about a fifth the diameter of this mass of probe droids. The Empire would have to have built this many in under three years. Can they do that?
Absolutely. Remember, these people rank high on the Kardashev scale. The second Death Star was 900 kilometers across. Four years after the destruction of the first, it was some significant fraction of the way completed. So in terms of sheer amount of mass manufactured, enough probe droids to send one to every planet in three years would not be a problem.
But they didn’t do that! They only sent thousands, not hundreds of billions. So their odds are back down to a couple chances in a million. If they have, say, 20,000 probe droids, then it’s one in nine million. It’s more likely than winning the lottery, but less likely than being dealt a royal flush in poker. They could have made enough droids to have an actual chance, but they didn’t. Clearly the problem is that Darth Vader isn’t obsessive enough. But he’ll get lucky anyway, because when movie makers get lazy, coincidences happen.
Never tell me the odds! —Darth Vader