>How many piano tuners are there in the entire world?
I can't see how there would be a correlation between answering this question well and being a good Product Manager.
What's most upsetting about this style of question is the 'holier than thou' attitude that comes along with them. If you're not willing to play the game, don't expect to be seen as a good candidate.
This question seems to be pulled straight from the classic 'Fermi estimate' [1]. However I have to say these are actually very useful in quickly assessing options given limited information. I remember doing something like this when discussing a claim about total information produced in history that another person felt was outrageous, I quickly threw together an estimate to show this wasn't outrageous and a physicist friend of mine pointed out the proper term for this. I can certainly see where the ability to do this on the spot would aid a product manager (and in this case I would say the ability to perform this estimate is independent of ever having heard the term 'fermi estimate')
The point of these sorts of questions is not to find people who know how many piano tuners there are in the world, or even to arrive at a reasonable estimate (without good data you're going to be off by miles).
The point is instead to find people who have developed the ability and habit of "back of the envelope" calculation. Not everyone has developed this skill but it can very much elevate the level of many technical discussions. Simply being able to come up with a guess that is within a factor of 2, 10, or even 1,000 (depending on context) can be critically important and impact decisions. Such crude calculations can also serve as the skeleton from which to create more robust calculations with better data. Enrico Fermi was able to estimate the yield of the first ever nuclear bomb test using only his eyes and a few pieces of torn paper blown by the blast, he came within a factor of 2. Sometimes you don't have the most precise measuring instruments or complete data, being able to cope with that and yet come up with useful results is also important.
I can't see how there would be a correlation between answering this question well and being a good Product Manager.
What if there's a product planned that specifically aims at piano tuners? Or a feature change that would affect piano tuners? What if Google Maps wants to add a "Find your local piano tuner" mode? Estimating how many potential users/customers there are, how to reach them and whether it's a significant segment is pretty useful. Sure, the example is bunk but it's meant to be something you have no clue about off the top of your head.
I can't see how there would be a correlation between answering this question well and being a good Product Manager.
What's most upsetting about this style of question is the 'holier than thou' attitude that comes along with them. If you're not willing to play the game, don't expect to be seen as a good candidate.