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. (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:

(a)

$\left[\begin{array}{ccccccccc} &1&2&3&4&5&6&7&8\\1&1&2&3&4&5&6&7&8\\2&2&4&6&8&10&12&14&16\\3&3&6&9&12&15&18&21&24\\4&4&8&12&16&20&24&28&32\\5&5&10&15&20&25&30&35&40\\6&6&12&18&24&30&36&42&48\\7&7&14&21&28&35&42&49&56\\8&8&16&24&32&40&48&56&64\end{array}\right]$

(b)

(i)probability that the product of the two numbers is a prime number=7/64

(ii)probability that the product of the two numbers is not a perfect square=56/64

(iii)probability that the product of the two numbers is a multiple of 5 =15/64

(iv)probability that the product of the two numbers is Less than or equal to 21 =40/64

(v) probability that the product of the two numbers is divisible by 4 or 6 =42/64

Step-by-step explanation:

(a) 8-sided fair die implies each die can have outcomes 1,2,3,4,5.6,7,8 with equal chances.

To draw the possibility diagram of the product of the two numbers appearing on the die in each throw, we put the outcomes of both dies side by side and find the product of possible outcomes.

(b) probability= Number of outcome / Total number of outcomes

(i)probability that the product of the two numbers is a prime number=8/64=1/8

based on sample space, the prime numbers are 2,2,3,3,5,5,7,7

(ii)probability that the product of the two numbers is not a perfect square=56/64=7/8

based on sample space, only  1,4,9,16,25,36,49,64 are perfect square

(iii)probability that the product of the two numbers is a multiple of 5 =15/64

(iv)probability that the product of the two numbers is Less than or equal to 21 =40/64=5/8

(v) probability that the product of the two numbers is divisible by 4 or 6 =42/64=21/32