Question

5. A six sided die is marked '2' on one face, “3' on two of its faces, and “4' on remaining three
faces. The die is thrown twice. If X denotes the total score in two throws, find the values of
the random variable and number of points in its inverse images.​

Answer 1

6.67 is the Mean Sum  of two throws

Step-by-step explanation:

Probability of  2  = 1/6

Probability of 3 = 2/6 = 1/3

Probability of 4 = 3/6  = 1/2

X = sum of throw in two Scores

= 4  ,  5  ,  6    ,  7  , 8

X = 4 = ( 2 + 2)  = (1/6)(1/6) = 1/36

X = 5 = (2 + 3) or (3 + 2)  =  (1/6)(1/3) + (1/3)(1/6)  =  1/9

X = 6 = (2 + 4) or (3 + 3) or ( 4 + 2)  =  (1/6)(1/2) + (1/3)(1/3) + (1/2)(1/6)  = 5/18

X = 7 =  (3 + 4)  or ( 4 + 3)  = (1/3)(1/2) + (1/2)(1/3)  =  1/3

X = 8  = ( 4 +  4)  = (1/2)(1/2)  = 1/4

X     Proabbility    

4      1/36

5      1/9

6      5/18

7      1/3

8      1/4

Mean = 1/9  + 5/9  + 5/3  + 7/3  + 2

= (1 + 5 + 15 + 21 + 18)/9

= 60/9

= 20/3

= 6.67 is the Mean Sum

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Answer 2

Answer:

Sample space is defined as the set of all possible outcomes.

when such a die is thrown once, the sample space is {2,3,4} only and not {2, 3, 3, 4,4,4} because we are only concerned about what falls and not which face. When there are only three numbers on the die, the only possible outcomes are 2,3,4.

So when it is thrown two times, there are only 9 possible combinations.

To say that the total is 36 is absolutely wrong.

If all the faces are differently coloured , then there will be  a distinction between white 3 or red 3 etc.

Therefore S={ (2,2), (2,3), (3,2), (2,4),(3,3) (3,4), (4,4), (4,3), (4,2) }

Hence X={4,5,6,7,8} and the inverse images are respectively {2,2)}, {(2,3) , (3,2)},  {(2,4), (3,3), (4,2)}, {(3,4), (4,3)}, {(4,4)}

Step-by-step explanation: Given above

The only thing is, each of the above outcomes is not equally likely.

P(X=4)=1/36,  P(X=5)=4/36,  P(X=6)=10/36, P(X=7)=12/36,  P(X=8)=9/36.