You are a student living in a small room with no closet.  All of your clothes sit on the floor. You can never remember if they are clean or dirty, and each day you have to decide whether to do the laundry or just throw something on.

If your strategy in these circumstance is to always do the laundry, you will be doing laundry every day, often washing clothes that are already clean.  On the other hand if your strategy is to dress and go your clothes will never get clean.

Instead you have to randomize.  If you wash with probability p then p is the probability you will be wearing clean clothes on any given day.  Of course you would like p=1, but then you are doing laundry with probability 1 every day.  Your optimal p is strictly between 0 and 1 and trades off the probability of clean clothes p versus the probability of washing clothes that are already clean.  (The latter is equal to the probability these clothes are clean, p, multiplied by the probability you wash them, again p so it’s p².)

The same logic applies to:

  1. I’ve been standing here in the shower for what must be a good 30 minutes; singing, sleeping, or absorbed in a proof that doesn’t work and I have forgotten whether I washed my hair. (shower cap nod:  David K. Levine)
  2. Its dark and I lost count of how many intersections I’ve crossed and I know I have to turn left somewhere to get home. (The classic example, due to Piccione and Rubinstein.)
  3. When was the last time I called my mother?
  4. etc…