Chapter 8.1, Problem 28E

Fitness A group of physical fitness devotees works out in the gym every day. The workouts vary from strenuous to moderate to light. When their exercise routine was recorded, the following observation was made: Of the people who work out strenuously on a particular day, 40% will work out strenuously on the next day and 60% will work out moderately. Of the people who work out moderately on a particular day, 50% will work out strenuously and 50% will work out lightly on the next day. Of the people working out lightly on a particular day, 30% will work out strenuously on the next day, 20% moderately, and 50% lightly.

a. Set up the $3\times 3$ stochastic matrix with columns and rows labeled S, M, and L that describes these transitions.

b. Suppose that, on a particular Monday, 80% have a strenuous, 10% a moderate, and 10% a light workout. What percent will have a strenuous workout on Wednesday?