Question 2 A dodecahedron is a regular polyhedron with 12 pentagonal faces. It is one of the 5 possible 3-dimensional shapes where every face is the same and every corner looks the same, known as a Platonic solid. The other Platonic Solids are 4 sided (tetrahedron), 6 sided (cube), 8 sided (octahedron) and 20 sided (icosahedron). A dice was made from a dodecahedron, and the faces were numbered 1 to 12. This dice was rolled once. We call the number from a 12-sided dice a "d12". (a) Calculate the mean and variance of the d12. (6 marks) (b) A game involves a board with a series of numbered squares (Numbers 1 to 200), and players take turns rolling a 12-sided die and moving their game piece forward along the squares based on the number rolled. For example, if a player rolls a 4, they move their piece forward 4 squares. In other words, this is a snake and ladder game without any complicated rules such as the snakes and ladders. Calculate the average number of d12 rolls needed to reach the 200th square. (c) X P(x) (d) (2 marks) We decided to change the rules of the game. Now, instead of d12, we will roll the d12 twice, and pick the larger number. For example, if we rolled 1 and 6, 6 will be selected. if we rolled 12 and 2, 12 will be selected. If we rolled 1 and 1, then 1 will be selected. Compute the probability of obtaining each of the numbers from 1 to 12 by filling up this table: 1 2 3 4 5 6 7 8 9 10 11 12 (6 marks) What is the average number of rolls needed to reach the 200th square under this ruleset? (6 marks)

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