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)