How Many Umbrellas Do You Need?

I’m on a blog break this week so all posts are reruns until the new year. This post originally appeared on 15 July 2009. If you wish to leave comments, please do so on the original post.

Rain was expected by the afternoon. I was not carrying an umbrella on my morning commute, and I had none at my office. So I purchased one on my way into work. It came with a “lifetime limited guarantee.” Does this suggest an improvement to the two-umbrella system described in the “umbrella problem” that was posed to me over a dozen years ago?

In 1996 I took an oral exam to qualify for my school’s PhD program. The exam included one question from each member of a three-professor panel of inquisitors. Professor Dimitri Bertsekas was on my panel, and he challenged me with ”the umbrella problem.” In those days neither the umbrella problem nor its solution was not on the internet. Today you can read about it in a Mathworks.com post and in a talk by Jeff Buzen (Power Point). In Example 6.5 (Chapter 6, page 15) of the Introduction to Probability lecture notes (co-authored by John Tsitsiklis), Bertsekas poses the umbrella problem as follows.

An absent-minded professor has two umbrellas that she uses when commuting from home to office and back. If it rains and an umbrella is available in her location, she takes it. If it is not raining, she always forgets to take an umbrella. Suppose that it rains with probability p each time she commutes, independently of other times. What is the steady-state probability she gets wet on a given day?

Unless you are accustomed to such problems it may not be immediately clear how to solve it. In the context in which it is posed in Bertsekas’ (and Tsitsiklis’) lecture notes it is simple. It is a straight forward application of basic properties of Markov chain models, which are described in the chapter that includes the problem.

So much of success in life is being prepared. I knew which professors would be on my oral exam panel. In the month before the exam I asked older graduate students what those same professors had asked prior years’ candidates. There was no guarantee that a professor would ask the same question twice. But this was a way to study questions likely to be similar to ones asked of me. Sure enough, the umbrella problem was a Bertsekas favorite; he asked it in prior years. I was prepared for it. I must have done well enough on the other two questions too, neither of which I recall, because I was admitted to the PhD program.

I think about the umbrella problem almost every rainy work day. In fact, it is probably due to the umbrella problem that I’ve implemented a two-umbrella system so I don’t always need to bring an umbrella with me if rain is expected later. But today for the first time, I forgot that the state of the system was zero umbrellas at work and two at home. I had forgotten to carry an umbrella from home and there was a good chance it would rain during my afternoon commute. Perhaps two umbrellas were not enough.

Hence my morning purchase of a Weather Zone brand “Oversize Automatic” black umbrella, style 1101, \$5.24 at 7-11. This bumped me from a two-umbrella system to a three-umbrella one. That it included a “lifetime limited guarantee” gave me an idea. If it ever breaks, threatening to disrupt my now three-umbrella system I could make a claim under the guarantee and get a new umbrella. Perfect!

Not so fast. It is a lousy guarantee. It only covers manufacturing defects of the umbrella frame, not the fabric, case, or handle and not “any damage caused by accident, abuse, or failure to allow the umbrella to dry in an open position.” Worse, to make a claim one must ship the defective umbrella, an original receipt, and \$5 (for shipping and handling) and wait four to six weeks for a repaired umbrella or replacement. Since paying to ship the umbrella would cost more than \$0.24 it would be far easier and less expensive to buy a new umbrella if my new one breaks.

But is it worth it? If my new umbrella breaks and I switch back to a two-umbrella system, what are the chances it will rain and I’ll be without an umbrella? Following the techniques in Bertsekas’ and Tsitsiklis’ lecture notes, it is no more than p*(1-p)/(3-p) where p is the probability of rain on my commute. The chances of rain in a day is about 0.34 where I live. Thus, the probability I’ll get wet is no greater than 0.08. It is almost certainly lower than 0.08 because the chances it will rain during my commute are lower than the chances it will rain in a given day. Plus I am not (yet) totally absent-minded. The solution to the umbrella problem shows that a two-umbrella system is quite good. The useless “limited lifetime guarantee” of my new umbrella does not offer an improvement.