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

(a) Shown below.

(b) Explained below.

Step-by-step 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.

25

step-by-step explanation:

median is the value separating the higher half of a data sample, a population or probablity distribution from the lower half.

step-by-step explanation:

say you have 3,8,7,3,1,9,8 order them from lowest to highest

if orderd it would be 1,3,3,7,8,8,9 now you take away from the lowest and the highest until you end up in the middle.

so you would remove the 1 and 9 leaving you with 5 digits remaining

when you've reached the middle that would be your answer so in this case the answer is 7

if there were two numbers left say 7 and 5 you'd and them together so 12 then divide by 2 witch would equal 6 so the answer would be 6

good luck on that project hoped i you at all