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:
- 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)
- 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.)
- When was the last time I called my mother?
- etc…
Cheap Talk 
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February 14, 2010 at 7:36 pm
NE1
My solution is to increase the number of grades in the triage. Clean clothes p=1 go in the dresser, right out of the dryer. Clothes I’ve worn once hang in the closet. Clothes I cannot wear without washing go in the hamper.
February 15, 2010 at 1:25 am
Lones Smith
Not simply to seize upon a great pun, but I think the best solution to your laundry solution is to purify your mixture: Sniff the shirt armpits quickly, and if they fail some mildly informative measure of nasal hot-or-not, toss it into the laundry. As long as the signal is continuous and ever so-slightly informative, it works. And I think it is a better descriptive of how the world works. Eg. in the absent-minded driver, look for that weak but fuzzy inner sense of deja vu. In the shower, why not feel your hair first?
Pondering such purification arguments have made me lose my faith in mixing. How can I get my mixed groove back? 😦
February 15, 2010 at 8:36 am
Ro'i
There is an implicit assumption that you have only one set of clothes, which renders the question of absentmindedness moot. Since you wear clothes every day, than you know that your clothes are dirty today, regardless of whether the were washed yesterday.
If you have more than one set of clothes, than the analysis is incorrect. The probability of wearing clean clothes becomes strictly larger than p, as you may be wearing clothes that were washed some time ago but not worn since. For example, if you assume two sets of clothes, with one chosen with equal probabilities each day, the probability of wearing clean clothes on a given day is the sum of [(1-p)^i*p*0.5^i], where i goes to infinity. The first argument is the probability of having last washed the clothes i days ago, and the second is the probability that you wore the same set of clothes in all of the days since the last wash, and that you wear the other set today.
February 15, 2010 at 6:44 pm
jeff
yes, thank you.
February 15, 2010 at 9:36 am
The Wife
Sniffing is out!!! I grab whatever Jeff left on the floor and wash them! 🙂 About the driving… that is why I drive instead of Jeff. Also Jeff only knows the road to and from work. OK… with a slight deviation from his driving path, he does know where to get coffee beans and where to get his car wash. As to his mother, he calls when he is waiting at the airport for his flight.
February 15, 2010 at 10:27 am
Vinnie
Take a whiff and check for stains–always the optimal strategy.
February 15, 2010 at 10:34 am
Alicia
For Laundry
1. in dresser or waiting to be folded on laundry table= clean
2.on top of dresser= been worn at least once but could be worn again
3. floor or hamper= dirty must wash
For directions in dark
1. turn on your lights
2. consider a gps system
Re calling parents
1. Set a time I call every saturday morning and then they do not worry or nag