{F} How many ways can we get an even sum when two distinguishable dice are rolled ?

The sample space is,

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

This sample space shows the outcome on the first dice followed by the outcome on the second dice.

Let the random variable $Z$ represent the sum of the outcome on the first dice and the outcome on the second dice.

Taking the sum of the outcome on the first and second dice gives,

$Z=\begin{Bmatrix} 2 & 3&4&5&6&7 \\ 3 & 4&5&6&7&8\\ 4&5&6&7&8&9\\ 5&6&7&8&9&10\\ 6&7&8&9&10&11\\ 7&8&9&10&11&12 \end{Bmatrix}$

From the values of the random variable $Z$ above, the number of times the values is even is 18.

Therefore, the number of ways we can get an even sum is 18 ways.