Question

Each morning an individual leaves his house and goes for a run. he is equally likely to leave either from his front or back door. upon leaving the house, he chooses a pair of running shoes (or goes running barefoot if there are no shoes at the door from which he departed). on his return he is equally likely to enter, and leave his running shoes, either by the front or back door. if he owns a total of k pairs of running shoes, what proportion of the time does he run barefooted?

Answer 1

The proportion of the time he runs barefooted is 1/(k+1) if he owns a total of k pairs of shoes. This is because a transition matrix shows the probabilities using Markov chains and these chains gives us the proportion.