Block Time Paradox

Imagine if the bitcoin block chain had only existed for 20 minutes and had found three blocks, two at time zero and one at time 20.

The average interval between blocks is 10 minutes. The first two blocks have an interval of zero. The second and third block have an interval of 20. The average is 10.

If we pick a random time, what is the average time to the previous block? At any point in time, the previous block is the second block. The time from that block runs from 0 to 20, so the average is 10.

If we pick a random time, what is the average time to the next block? At any point in time, the next block is the third block. The time to that block runs from 20 to 0, so the average is 10.

The explanation for this paradox is that we are averaging over different things. When we say the average time between blocks is 10 minutes, we are averaging over blocks, giving each block equal weight. When we say the average time to the next (or from the previous) block is 10 minutes, we are averaging over time, giving each minute equal weight.

When blocks are found in quick succession, they account for very few minutes. When blocks are found far apart, the account for very many minutes. So when we average over time, blocks found far apart get more weight than blocks found closer together.

This same phenomenon occurs in many other areas. The average bus driver will think buses are less crowded than the average bus rider. Why? Because riders are more likely to be on crowded buses and so will give them more weight. Drivers are equally likely to be on crowded or non-crowded buses. So they will give crowded buses equal weight.

Imagine two buses, each with one driver, one with ten passengers and one with one hundred passengers. The average driver sees 55 passengers per bus. The average passenger sees 92 or so. If this pattern repeats roughly the same over time, bus drivers will think the average bus has about 55 passengers while passengers will think the average bus has about 92 passengers. They are both right, they are just measuring different things.