"Assume a hashrate and difficulty corresponding to 1 block per 10 minutes. If I uniformly randomly pick a point in time, what is the expected time between the previous block and the next block?"
A more interesting question, I think, is given a POW algorithm which adjusts difficultly to target a 10 minute block rate, and assuming network hash rate is constant and propagation is instant, what percentage of blocks take 20 minutes to solve? 60 minutes? And what’s the odds of solving two blocks in <= 60 seconds?
* If I choose a time uniformly what is the expected time it will take to mine that particular block.
The difference between them being that blocks that take a long time occupy more space on the timeline than shorter blocks and are therefore more likely to be chosen.
How do you get this question from CyrusL's original question? CyrusL explicitly specifies that it is 10 minutes per block [0], and that he is uniformly picking a point in time.
[0] Admittedly, he does not specify that it is a Poisson distribution.