Zarkon. Attractive land of exotic flora and fauna. Planet with many historical sites relating to the transcendental explosion of 3042. World of marvelous cities filled with delightful gift boutiques. Zarkon, home to the most unintuitive bus system in the galaxy.

You, an intrepid space tourist, have stranded yourself here after hopping on the wrong pleasure barge at planet Chlorox. This is embarrassing. You don’t even speak Zarkonese.

Walking about the city in an effort to find a minimally exotic bar and a place to stay for the night, you discover a bus stop. You look around the stop for a schedule, even though you can’t read Zarkonese. There isn’t a schedule anyway. Thankfully, a bizarre, one-headed creature waiting at the stop happens to speak your language. It informs you that the busses do not travel on a regular schedule. "Oh. Well, has it been a long time since the last one came?", you ask.

"It doesn’t matter, the arrival of busses is a memoryless stochastic process," the creature replies. "The arrival times of past busses don’t tell you anything about when the next one will come. However, the average time between bus arrivals is 40 zinutes."

You mentally process this for a moment. "So I guess one could come right now, or maybe not until 5 zours from now," you say. "How long should I expect to wait for one, on average?"

"You tell me. I’ve never been any good at math."

You begin to think about the problem. "Hmm. How long do I need to wait? I suppose 40 zinutes. Wait, that doesn’t seem right. The busses are an average of 40 zinutes apart, but I could come to the stop at any point between them. It would seem that, on average, I’d come to the stop halfway between two bus arrivals. So maybe I should expect to wait 20 zinutes. But does that really fit with the bus arrivals being memoryless? If I come 20 zinutes after a bus arrival I should expect to wait just as long as if I came immediately after a bus arrival. Gee, now I’m really not sure what the answer is."

How long do you have to wait, on average? The answer is rot13’d below.

Gur rkcrpgrq jnvgvat gvzr vf sbegl mvahgrf. Guvf fbzrjung pbhagrevaghvgvir snpg vf n jryy-xabja cnenqbk / fhecevfvat snpg jvgu fgbpunfgvp cebprffrf. Fb jul vf gur rkcrpgrq jnvgvat gvzr sbegl mvahgrf?

Vg vf orpnhfr lbh graq gb neevir ng gur ohf fgbc qhevat bar bs gur ybatre vagreinyf. Gur nirentr yratgu bs gur vagreinyf orgjrra ohf neevinyf vf sbegl mvahgrf, ohg vs lbh cvpx n enaqbz gvzr, gur nirentr yratgu bs gur vagreiny gung pbagnvaf gung gvzr vf npghnyyl rvtugl mvahgrf. Sebz guvf cbvag, gur hfhny ernfbavat gung fhttrfgf lbh neevir va gur zvqqyr bs gur vagreiny nccyvrf, tvivat n svany erfhyg bs sbegl mvahgrf.

Gung jnf na vaghvgvir rkcynangvba; lbh pna qrevir vg zngurzngvpnyyl. Gur snpg gung gur ohf neeviny cebprff vf zrzbelyrff vzcyvrf gung vs lbh gnxr gur gvzr ng juvpu gur arkg ohf neevirf gb or n enaqbz inevnoyr, vgf cebonovyvgl qrafvgl shapgvba vf

Lbh pna jbex bhg gur erfhyg sebz gurer.