Question

An 8-sided fair die is rolled twice and the product of the two numbers obtained when the die is rolled two times is calculated.

(a) Draw the possibility diagram of the product of the two numbers appearing on the die in each throw?

Answer 1

Question Completion

(b) Use the possibility diagram to calculate the probability that the product of the two numbers is

I) A prime number

ii) Not a perfect square

iii) A multiple of 5

iv) Less than or equal to 21

v) Divisible by 4 or 6

Answer:

I) 0.125 (ii)0.828125 (iii)0.234375 (iv) 0.625 (v)0.65625

Explanation:

The sample space for an 8-sided fair die rolled twice is:

`[(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),(1, 7),(1, 8)\\(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),(2, 7),(2, 8)\\(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),(3, 7),(3, 8)\\(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),(4, 7),(4, 8)\\(5, 1), (5, 2), (5, 3), (5, 4), (5, 5),(5, 6),(5, 7),(5, 8)\\(6, 1), (6, 2), (6, 3), (6, 4)(6, 5),(6, 6),(6, 7),(6, 8)\\(7, 1), (7, 2), (7, 3), (7, 4)(7, 5),(7, 6),(7, 7),(7, 8)\\(8, 1), (8, 2), (8, 3), (8, 4)(8, 5),(8, 6),(8, 7),(8, 8)]\\`

(a) The possibility diagram of the product of the two numbers appearing on the die in each throw

`1, 2, 3, 4, 5, 6,7,8\\2, 4, 6, 8, 10, 12,14,16\\3,6,9,12,15,18,21,24\\4,8,12,16,20,24,28,32\\5,10,15,20,25,30,35,40\\6,12,18,24,30,36,42,48\\7,14,21,28,35,42,49,56\\8,16,24,32,40,48,56,64`

(b)

I) A prime number

The number of products that gives prime numbers = 8

`P$(Product is a prime number)=(8)/(64)= (1)/(8)\\=0.125`

ii) Not a perfect square

Number of products which results in perfect squares =11

`P$(Product is Not a perfect square)=(64-11)/(64)= (53)/(64)\\=0.828125`

iii) A multiple of 5

Number of products that are multiples of 5=15

`P$(product is a multiple of 5)=(15)/(64)\\=0.234375`

iv) Less than or equal to 21

Number of products which are less than or equal to 21=40

`P$(product is less than or equal to 21)=(40)/(64)=(5)/(8)\\=0.625`

v) Divisible by 4 or 6

Products Divisible by 4 or 6= 42

`P$(product is divisible by 4 or 6)=(42)/(64)=(21)/(32)\\=0.65625`