question 2 a dodecahedron is a regular polyhedron with 12 pentagonal f

Question

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)