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
Answers
Given : 8-sided fair die is rolled twice and the product of the two numbers obtained
To find : possibility of the product of the two numbers appearing on the die in each throw
Solution:
8-sided fair Dice has output = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 }
Dice is thrown twice so possible outputs are 8 * 8 = 64
= { (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) }
Products
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)
24 = ( 3 , 8) , ( 4 , 6 ) , ( 6 , 4) , ( 8 , 3)
25 = ( 5 , 5)
28 = ( 4 , 7) , ( 7 , 4)
30 = ( 5 , 6) , ( 6 ,5 )
32 = ( 4 , 8) , ( 8 , 4)
35 = ( 5 , 7) , ( 7 , 5)
36 = (6 , 6)
40 = ( 5 , 8) , ( 8 , 5)
42 = ( 6 , 7) , ( 7 , 6)
48 = ( 6 , 8) , ( 8 , 6)
49 = ( 7 , 7)
56 = ( 7 , 8) , ( 8 , 8)
64 = ( 8 , 8)
Products = { 1 ,2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 ,12 , 14 , 15 , 16 , 18 , 20 ,21 , 24 , 25 , 28 , 30 ,32 , 35 , 36 , 40 , 42 , 48 , 49 , 56 , 64 }
product of the two numbers is (i) A prime number 2 , 3 , 5 , 7
8 Possible Case
Probability = 8/64 = 1/8
(ii) Not a perfect square = 1 - perfect Square
Perfect Square 1 , 4 , 9 , 16 , 25 , 36 , 49 , 64
possible Cases = 12
Probability = 1 - 12/64 = 1- 3/16 = 13/16
(iii) A multiple of 5 = 5 , 10 , 15 , 20 , 25 , 30 , 35 , 40 ,
15 possible Cases
Probability = 15/64
(iv) Less than or equal to 21
possible cases = 40
Probability = 40/64 = 5/8
(v) Divisible by 4 or 6 , 4 , 6 , 8 , 12 , 16 , 18 , 20 , 24 , 28 , 30 , 32 , 36 , 40 , 42 , 48 , 56 , 64
42 possible cases
Probability = 42/64 = 21/32
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