A probabilist has five pairs of jogging shoes, which he never takes inside his house, but leaves just outside the front door or the back door before he enters his house. Each evening, he randomly walks out of either door. If there are no shoes there, he walks barefoot in his own yard before entering home again. Otherwise, he puts on a pair of shoes, goes jogging in the neighborhood, and on his return he enters through either door chosen at random after taking off his shoes just outside that door. Assume that on Day 1, when he is about to go out, he has all five pairs of shoes at the front door. On Day 366, how many shoes are there at the front door? The answer is not a number, but a random variable. Give its probability mass function. During Days 1 through 365, how many days did he walk barefoot in his yard?