The second sentenxe doesn't follow the first - it's a flavor of Zeno's paradox, after 5 years, you'll be left with 4/5 x 4/5 x 4/5 x 4/5 x 4/5 of the original batch.
You made a mistake: it's not that 1/5 of the computers spontaneously break every year. It's that 1/5 of the students treat their computers roughly.
Assuming that laptops get collected over the summer and re-distributed each year, you should actually expect that 100% of each tranche of laptops would need to be replaced every 5 years.
That's fair. I guess, it's more accurate to say that you'd expect a number of laptops equal to the size of the initial tranche to be destroyed after the first 5 years.
Although if I was running IT, it'd definitely keep track of the "destructive" students and issue them the oldest equipment, in which case, we'd be back to something closer to my original statement.
And anyway, if 1/5 of laptops needed major repairs and some of them got it, those go back into circulation. Are they still original laptops? (Ask my grandfather's axe...)
It's not the case that there's a constant 1/5 probability of failure each year. Many failure modes are based on cumulative stress/degradation; so the probability of failure can go up over time.
Some failure modes go down over time; maybe there's some manufacturing defect, and those that have the defect fail early, while those that survive past the first year will have lower chances of failure early on.
But in this kind of environment, the cumulative stresses are much more likely than the early failures.
Anyway, it doesn't matter much either way, even if there's a few that survive, they will be having wear and tear to the point you wouldn't want another student to have them (or maybe as a replacement for a broken one); you wouldn't want some year 1 students to get a new ones while others get the year 4/5 leftovers, they'll resent it for sure.
Second, they'll be paid off after a few years.
You absolutely could design it to last 20 years with batteries that are easy to pop out and pop in as easy as changing a double aa but your customers won't pay a premium over a more disposable machine and indeed if your customer has a good experience over the 3-5 they actually use it for you make MORE if your hardware is designed to need replacement.