Question 

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.

(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.