You went astray when you declared the expected wait and expected passed.
Draw a number line. Mark it at intervals of 10. Uniformly randomly select a point on that line. The expected average wait and passed (ie forward and reverse directions) are both 5, not 10. The range is 0 to 10.
When you randomize the event occurrences but maintain the interval as an average you change the range maximum and the overall distribution across the range but not the expected average values.
When you randomize the event occurences, you create intervals that are shorter and longer than average, so that a random point is more likely to be in a longer interval, so that the expected length of the interval containing a random point is greater than the expected length of a random interval.
To see this, consider just two intervals of length x and 2-x, i.e. 1 on average. A random point is in the first interval x/2 of the time and in the second one the other 1-x/2 of the time, so the expected length of the interval containing a random point is x/2 * x + (1-x/2) * (2-x) = x² - 2x + 2, which is 1 for x = 1 but larger everywhere else, reaching 2 for x = 0 or 2.
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.
You went astray when you declared the expected wait and expected passed.
Draw a number line. Mark it at intervals of 10. Uniformly randomly select a point on that line. The expected average wait and passed (ie forward and reverse directions) are both 5, not 10. The range is 0 to 10.
When you randomize the event occurrences but maintain the interval as an average you change the range maximum and the overall distribution across the range but not the expected average values.