Question 1. J oJo Mobile sells data plans for cellphones. They sell Individual Plans as well as Family Plans. Market analysts at JoJo Mobile have noticed that customers tend to switch between the two plans each year. 0 Among the customers with an individual plan, 30% switch to a family plan the next year. 0 Among the customers with a family plan, 20% switch to an individual plan in the next year. This information can be encoded into a 2 X 2 matrix, as follows: 0.? 0.2 P {0.3 as} 400 1 (a) Compute the vector P [500 , and give a physical interpretation of what this vector represents. 740 3 (b) The vector P [550 , and give a physical interpretation of what this vector represents. (c) An eigenvector for P with eigenvalue A = l, and give a physical interpretation of what this vector represents. (d) An eigenvector for P with eigenvalue A c: 1. Explain why the eigenvector does not have a physical interpretation. Question 2. A standard deck of fty-two cards has thirteen ranks R = {A,2, 3,4,5, G,7,8,9, ll}, J,Q,K} and four suits S = {U, 0, Qt, O}. In MAT 133, we play cards with an extended deck of 70 cards. This deck has an additional suit of cards: the smile suit ; and an additional rank: the Professor. For example, the "professor of smiles" card is P. The ranks in this extended deck, in consecutive order, are R = {14, 2, 3, 4, 5, G, 7, 8,9,10,J,Q, K, P} Using this extended deck, count the number of ways to form each of the following hands: (a) The number of six-card hands that contain exactly two Professors and exactly three smiles. (Example: P? P. 8 Q. J Q is a valid hand.) (b) The number of six-card hands that contain one three-of-a-kind and three other cards that have consecutive ranks. (Example: 3' 3% 3. 8@ 9" 10% is a valid hand.)