This puzzle was originally written by Grue. I found a version of it on Lawrence Kesteloot's homepage.

You have just started working at a new company. At this company, there are two corkboards hanging on the wall. Each employee of the company has their photo tacked to one of the boards. The photos are ordered by seniority (that is, how long the employee has been with the company) with the most senior employees on the left board and the least senior employees on the right board. Only current employees have photos on the board. Each board fits 150 photos, and they are both full. Since you just started working for the company, your photo is in the last spot on the right board. You wonder how long it will take before your photo is on the left board.

Suppose that each week one person quits and another person is hired. The person who quits is random with each current employee being equally likely to quit. Assuming you don't quit, how long will it take, on average, before you are on the first board?

Here is the answer.