Logic Puzzles vs. Gambler’s Paradox

I got this one from Alasdair months ago, and it’s a doozy. However, it’s really only intended for the mathematically minded, so feel free to skip if you’re not comfortable with introductory probability. It’s a multi-part question, but the first parts are intended to be easy, so don’t second-guess yourself too much on those.

  1. A guy is repeatedly rolling an unbiased six-sided die in a room when you enter the room and start watching him. How many rolls do you expect to watch before he rolls a 1?
  2. On further inspection, you notice that he has written down his previous rolls in a list. How far back in the list do you expect to have to look before seeing the previous 1? You can assume that the guy has been doing this for long enough that he rolled a 1 at some point in the past.
  3. Define a “drought of 1’s” as the die rolls between successive 1’s (including the bounding 1’s). For instance, the rolls 1-4-2-1-5-1 would constitute a drought of length 4 followed by a drought of length 3. What is the expected length of a drought of 1’s?

The typical answers to the above questions are six, six, and seven (since we’re including both the starting 1 and the ending 1). However, this implies that on average droughts will have length seven, but when you walk into the room the average drought will have length twelve: you’d expect to be six rolls into the drought already, and you’d expect six more once you start watching. What’s going on here? How should this discrepancy be resolved?

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