Answer:
(a) Shown below.
(b) Explained below.
Explanation:
(a)
The sample space of rolling an 8-sided die twice is as follows:
S = {(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)}
There are a total of N = 64 elements.
(b)
(i)
The product of the two numbers is a prime number:
Product is a prime number samples:
2 = ( 1, 2) , ( 2, 1)
3 = ( 1 , 3) , ( 3 , 1)
5 = ( 1, 5) , ( 5 , 1)
7 = ( 1, 7) , ( 7 , 1)
Number of samples, n = 8
P (Product is a prime number) = 8/64 = 1/8 = 0.125.
(ii)
The product of the two numbers is not a perfect square :
Product is not a perfect square samples:
2 = ( 1, 2) , ( 2, 1)
3 = ( 1 , 3) , ( 3 , 1)
5 = ( 1, 5) , ( 5 , 1)
6 = ( 1, 6) , ( 2, 3) , ( 3, 2) , ( 6 , 1)
7 = ( 1, 7) , ( 7 , 1)
8 = ( 1 , 8) , ( 2, 4) , ( 4 , 2) , ( 8 , 1)
10 = ( 2, 5) , ( 5, 2)
12 = (2 , 6) , ( 3 , 4) , ( 4 3) , ( 6 , 2)
14 = ( 2, 7) , ( 7 , 2)
15 = (3 , 5) , ( 5 , 3)
18 = ( 3, 6) , ( 6 , 3)
20 = ( 4, 5) , ( 5, 4)
21 = ( 3 , 7) , ( 7 , 3)
24 = ( 3 , 8) , ( 4 , 6 ) , ( 6 , 4) , ( 8 , 3)
28 = ( 4 , 7) , ( 7 , 4)
30 = ( 5 , 6) , ( 6 ,5 )
32 = ( 4 , 8) , ( 8 , 4)
35 = ( 5 , 7) , ( 7 , 5)
40 = ( 5 , 8) , ( 8 , 5)
42 = ( 6 , 7) , ( 7 , 6)
48 = ( 6 , 8) , ( 8 , 6)
56 = ( 7 , 8) , ( 8 , 8)
Number of samples, n = 52
P (Product is not a perfect square) = 52/64 = 0.8125
(iii)
The product of the two numbers is a multiple of 5:
Product is a multiple of 5 samples:
5 = ( 1, 5) , ( 5 , 1)
10 = ( 2, 5) , ( 5, 2)
15 = (3 , 5) , ( 5 , 3)
20 = ( 4, 5) , ( 5, 4)
25 = ( 5 , 5)
30 = ( 5 , 6) , ( 6 ,5 )
35 = ( 5 , 7) , ( 7 , 5)
40 = ( 5 , 8) , ( 8 , 5)
Number of samples, n = 15
P (Product is a multiple of 5 ) = 15/64 = 0.2344.
(iv)
The product of the two numbers is less than or equal to 21:
Product is less than or equal to 21 samples:
1 = ( 1, 1)
2 = ( 1, 2) , ( 2, 1)
3 = ( 1 , 3) , ( 3 , 1)
4 = (1 , 4) , ( 2, 2) , ( 4, 1)
5 = ( 1, 5) , ( 5 , 1)
6 = ( 1, 6) , ( 2, 3) , ( 3, 2) , ( 6 , 1)
7 = ( 1, 7) , ( 7 , 1)
8 = ( 1 , 8) , ( 2, 4) , ( 4 , 2) , ( 8 , 1)
9 = ( 3, 3)
10 = ( 2, 5) , ( 5, 2)
12 = (2 , 6) , ( 3 , 4) , ( 4 3) , ( 6 , 2)
14 = ( 2, 7) , ( 7 , 2)
15 = (3 , 5) , ( 5 , 3)
16 = (2 , 8) , ( 4 , 4) , ( 8 , 2)
18 = ( 3, 6) , ( 6 , 3)
20 = ( 4, 5) , ( 5, 4)
21 = ( 3 , 7) , ( 7 , 3)
Number of samples, n = 40
P (Product is less than or equal to 21) = 40/64 = 0.625.
(v)
The product of the two numbers is divisible by 4 or 6:
Product is divisible by 4 or 6 samples:
4 = (1 , 4) , ( 2, 2) , ( 4, 1)
6 = ( 1, 6) , ( 2, 3) , ( 3, 2) , ( 6 , 1)
8 = ( 1 , 8) , ( 2, 4) , ( 4 , 2) , ( 8 , 1)
12 = (2 , 6) , ( 3 , 4) , ( 4 3) , ( 6 , 2)
16 = (2 , 8) , ( 4 , 4) , ( 8 , 2)
18 = ( 3, 6) , ( 6 , 3)
20 = ( 4, 5) , ( 5, 4)
24 = ( 3 , 8) , ( 4 , 6 ) , ( 6 , 4) , ( 8 , 3)
28 = ( 4 , 7) , ( 7 , 4)
30 = ( 5 , 6) , ( 6 ,5 )
32 = ( 4 , 8) , ( 8 , 4)
36 = (6 , 6)
40 = ( 5 , 8) , ( 8 , 5)
42 = ( 6 , 7) , ( 7 , 6)
48 = ( 6 , 8) , ( 8 , 6)
56 = ( 7 , 8) , ( 8 , 8)
64 = ( 8 , 8)
Number of samples, n = 42
P (Product is less than or equal to 21) = 42/64 = 0.6563.