Question

A green die and a red die are rolled. The random variable X is the absolute value of the difference between the number of spots on the red die and green die.

Answer 1

The number of spots on the red die and green die is given in the sample space $S$ below. The first value is the number of spots on the red die and the second value is the number of spots on the green die.

$S=\begin{Bmatrix} (1,1) & (2,1)&(3,1)&(4,1)&(5,1)&(6,1) \\ (1,2) & (2,2)&(3,2)&(4,2)&(5,2)&(6,2)\\ (1,3)&(2,3)&(3,3)&(4,3)&(5,3)&(6,3)\\ (1,4)&(2,4)&(3,4)&(4,4)&(5,4)&(6,4)\\ (1,5)&(2,5)&(3,5)&(4,5)&(5,5)&(6,5)\\ (1,6)&(2,6)&(3,6)&(4,6)&(5,6)&(6,6) \end{Bmatrix}$

From the sample space, the absolute values of the difference between the number of spots on the red die and green die is given below.

$\begin{Bmatrix} (0) & (1)&(2)&(3)&(4)&(5) \\ (1) & (0)&(1)&(2)&(3)&(4)\\ (2)&(1)&(0)&(1)&(2)&(3)\\ (3)&(2)&(1)&(0)&(1)&(2)\\ (4)&(3)&(2)&(1)&(0)&(1)\\ (5)&(4)&(3)&(2)&(1)&(0) \end{Bmatrix}$

Clearly,

there are 6 out out 36 values whose absolute difference is 0.

there are 10 out of 36 values whose absolute difference is 1.

there are 8 out of 36 values whose absolute difference is 2.

there are 6 out of 36 values whose absolute difference is 3.

there are 4 out of 36 values whose absolute difference is 4.

there are 2 out of 36 values whose absolute difference is 5.

Now, let the random variable $X$represent the absolute value of the difference between the number of spots on the red die and green die. The random variable $X$ may take on the following values.

$x=0,1,2,3,4,5$

The probability distribution of $X$ is given below.

$x$ 0 1 2 3 4 5

$p(X=x)$ ${6\over36}$ ${10\over36}$ ${8\over36}$ ${6\over36}$ ${4\over36}$ $2\over36$