A random question. At the very first try we succeeded. Though it might not be strange, What was the probability of surviving so long with so many external variables and constants.?
It can't be pure luck, seeing as how both Voyager spacecraft are still operating today. If it were just one, you might chalk it up to luck. Both still working means they must have been built really well. Which they were.
It is essential of course that they are nuclear powered, and also that we have really big antennas on Earth with which to talk to them. And that there was attitude control fuel leftover after their planetary flybys. But none of those are luck.
Things have failed on them, mostly moving parts and some capacitors, but we have found ways to work around all of the problems. Again, not luck.
A thing works fine until something goes wrong.
As far as I can see, the main external risks of the voyage were the radiation belts of Jupiter and Saturn, but those were already survived by the Pioneer 10 and 11 missions. After the planetary fly-bys, there are close to nothing external affecting the spacecraft. The important risks therefore lies in the potential failure of of one of the spacecraft’s subsystems:
The power source of the Voyager missions was an RTG. This requires no moving parts, and the decay of radioactive isotopes is not dependent on any external factors. This will probably work just fine until the power level has decreased enough.
A computer has multiple potential weaknesses, the most common ones being overheating and memory malfunction, like what happened on the Galileo mission. Radiation in outer space will over time slowly degrade electronic components. Magnetic memory will over time degenerate anyway.
It seems like what eventually will likely cause the Voyagers to fail is their nearly four decade old electronics.
The probes needed propellant for changing attitude, as well as for trajectory corrections. Corrosive propellant staying in the tanks for years are a major cause of spacecraft failures. See for instance the Akatsuki probe.
The Voyagers' longevity is not due to luck.
The Voyager program really began with the Pioneer 10 and 11 missions. They provided essential data on the conditions the Voyagers were likely to encounter, both in interplanetary space and in the vicinity of Jupiter and Saturn. The Pioneers were a huge jump for NASA, being the first interplanetary craft that had to operate for more than a few months.
The environment around Jupiter turned out to be the worst: it alone accounts for about 50% of the radiation dose throughout the Voyager missions.
A great deal of reliability analysis was done on the Voyager spacecraft design, leading to (page 55 of long PDF) ...
Probability of success for the Voyager capsule bus preferred concept is estimated at 0.830.
Still, the Voyager probes have far exceeded expectations. Looking at engineers’ testimonials from when it was built, Dodd said the original designers were told not to worry about reaching interstellar space and focus on making sure the Voyagers could observe Jupiter and Saturn.
“Basically they kind of ignored those directions, nodded their heads and did what they wanted to make it capable of getting to interstellar space,” she said.
Over the years, there have been failures. The biggest problem was the scan platform getting stuck. That was reversible, though. As the power available drops due to decay, instruments have to be switched off.
There are some AAIA papers that provide more information, unfortunately they don't seem to be available online:
Due to reliability requirements placed on the Voyager spacecraft system design and a mission resulting in long two-way, light time communication links, on-board automatic fault detection and correction capabilities are a significant feature of that spacecraft's design. Most of the protection to otherwise mission-catastrophic failures is implemented in the software of the voyager's central computer, while some resides in an attitude control-dedicated processor.
Tests and analysis indicated the need to protect the Voyager spacecraft from electrostatic discharges in the Jupiter energetic charged particle environment. This paper describes the identified problem areas, the test and analysis methods and results, the resultant design changes, and the final assessment of the design changes and verification methods.
You give a good group of engineers enough time, money and resources to build a decent piece of equipment, and its going to last for a while. Proper practice is to overdesign for known stresses by a factor of three or more. That gives you leeway with unexpected stresses to. No proper engineer wants to see his creation fail because he bought the 29 cent capacitor instead of the $1.69 one.