I think I understand my mistake. As the variance of the intervals widens the average event interval remains the same but the expected average distances for a sample point change. (For some reason I thought that average distances wouldn't change. I'm not sure why.)
Your example illustrates it nicely. A more intuitive way of illustrating the math might be to suppose 1 event per 10 minutes but they always happen in pairs simultaneously (20 minute gap), or in triplets simultaneously (30 minute gap), or etc.
So effectively the earlier example that I replied to is the birthday paradox, with N people, sampling a day at random, and asking how far from a birthday you expect to be on either side.
If that counts as a paradox then so does the number of upvotes my reply received.
Your example illustrates it nicely. A more intuitive way of illustrating the math might be to suppose 1 event per 10 minutes but they always happen in pairs simultaneously (20 minute gap), or in triplets simultaneously (30 minute gap), or etc.
So effectively the earlier example that I replied to is the birthday paradox, with N people, sampling a day at random, and asking how far from a birthday you expect to be on either side.
If that counts as a paradox then so does the number of upvotes my reply received.