Question

Q4. 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
(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 1

Answer:

I) $\dfrac{1}{8}$  (ii) $\dfrac{53}{64}$   (iii) $\dfrac{15}{64}$   (iv) $\dfrac{5}{8}$   (v)$\dfrac{21}{32}$

Step-by-step 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) 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  

$P (Product is a prime number)=\dfrac{8}{64}= \dfrac{1}{8}$

ii) Not a perfect square  

Number of Perfect Squares =11

P(Product is Not a perfect square)  $=\dfrac{64-11}{64}= \dfrac{53}{64}$

iii) A multiple of 5

Number of Multiples of 5=15

$P (A multiple of 5)=\dfrac{15}{64}$

iv) Less than or equal to 21

Product less than or equal to 21=40

$P (Less than or equal to 21)=\dfrac{40}{64}=\dfrac{5}{8}$

v) Divisible by 4 or 6

Products Divisible by 4 or 6= 42

$P (Divisible by 4 or 6)=\dfrac{42}{64}=\dfrac{21}{32}$