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In a bus station the probability of a bus coming in the next minute is $1/10$.

If we define $X$ as a random variable who point at which minute the bus is coming, it's easy to see that $X$ is geometrically distributed with $p = 1/10$.

So we know that $E(X) = 10$.

Now say we got to the bus station on a random time, what is the expected value between the last bus that arrived and the next bus ( the one we are waiting for).

I'm a bit confused as on one hand it's seems it should be $10$, but on the other hand it's seems like it should be $20$...

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This is a discrete version of what's called the waiting time paradox. Given a set of bus arrival times for which the interarrival times aren't all the same, you're more likely to get to the bus stop during one of the longer interarrival times. So the expected value between the time the last bus arrived and when the next bus arrives will be more than 10. –  Mike Spivey Dec 17 '12 at 4:56

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