Suppose there are three brands of laundry detergent: A, B, and C. Let Xn be the brand chosen on the nth purchase, where the states are 0 (brand A), 1 (brand B), and 2 (brand C). Customers who try these brands are satisfied and chose the same thing again on the next purchase with probabilities 0.5, 0.7, and 0.3, respectively. The probability a customer will change brands on their next purchase to one of the other two brands is equally likely. Write the one-step transition probability matrix.

Luke started a weight-loss program. the first week, he lost x pounds. the second week, he lost pounds less than times the pounds he lost the first week. the third week, he lost 1 pound more than of the pounds he lost the first week. liam started a weight-loss program when luke did. the first week, he lost 1 pound less than times the pounds luke lost the first week. the second week, he lost 4 pounds less than times the pounds luke lost the first week. the third week, he lost pound more than times the pounds luke lost the first week.