Math 6010 • Linear Models

A thinker who owns \(r\) umbrellas travels back and forth between home and office, taking along an umbrella (if there is one at hand) in rain (probability \(p\)) but not otherwise (probability \(q=1-p\)). Let the state be the number of umbrellas at hand, irrespective of whether the thinker is at home or at work. Set up the transition matrix, and show that the Markov chain approaches equilibrium (i.e., the ergodic theorem is applicable). Find the steady-state probability of his getting wet, and show that five umbrellas will protect him at the \(5\%\) level against any climate (any \(p\)).