It's been reverse engineered by fiddling with the clock on android simulators. They found the point that was equal to 100% rare chance and worked backwards. 100% is at 25 days, but good luck getting that far.
Your math is off though on the shorter explore probabilities.
You're looking at 1- (.99)^9 which is 8.6% of rare. Similarly worse for legendaries as well.
The chance of legendaries increases with time too so the guy out at 54 hours has a 1% chance. While the short explorers are sitting at 1 - (.999)^9 = .89%
And lets not forget they get to *sustain* that increased rate for as long as they can last out there. And it increases with every hour.
If dwellers were only able to carry a limited amount of loot your strategy would actually not be bad based on the probabilities. But with infinite inventories it eventually falls behind because after that 54 hour mark my dwellers last another 6-12 hours and have 6-12 more chances at rare loot at the increased 10% rate. My likelihood of getting a rare at just 6 hours past the 54 hour mark is 46% at 12 hours it's 71%.
That's huge.
And I'm actually not recalculating the chances, they are higher than that since they increase with every hour so these are lowballs for my numbers. And it's only the probability of getting at least 1 rare.
I'm pretty convinced by the math that longer hauls are superior if you have the dwellers and resources to sustain them.
The best way is to send a fully trained lvl 1-9 dweller out with a fat man and the best endurance equipment you have.
Their level ups are free stimpacks on the road. They end up not using any stimpacks until the 2 day mark or 48 hours. From there it's a breeze to get past 54 hours and keep going for a long while.