Solution
The Waiting Time Paradox
Show the problem again
Buses arrive at a stop according to a Poisson process with an average of one bus every 10 minutes. You arrive at a uniformly random time. What is your expected wait for the next bus, in minutes?
Worked solution
The answer is 10 minutes, not 5. By the memorylessness of the exponential distribution, your wait from any arrival instant is exponential with mean 10, regardless of when the last bus came. The "5 minutes" intuition fails due to the inspection paradox: a random arrival time is more likely to land inside a long inter-bus gap than a short one, so the gap you sample is length-biased: its expected length is 20 minutes, and you wait half of it on average.
Source: The inspection paradox, treated in William Feller's 'An Introduction to Probability Theory and Its Applications'. Statement written for AxiomIQ.