Question

What is the probability that rolling four fair, six-sided dice produces an even sum?

Answer 1

even sum 2, 4, 6, 8 , 10, 12

ways 1,1 one way
3,1;2,2; 1,3 three ways
5,1;4,2;3,3;3,3;1,5;2;4 six ways
and continue...

So for the pr (2)=1/6*1/6
pr(4)=1/6*1/6*3
pr(6)=1/6*1/6*6
and continue

Answer 2

To have an even sum
1. all four must be even (E)
2. all four must be odd (O)
3 two must be even, two must be odd.

prob that one die lands even = 3/6 = 1/2
the same as for odd.

prob of EEEE = (1/2)^4
prob of OOOO = (1/2)^4
prob of EEOO = 4!/(2!2!) (1/2)^4

sum = 1/16 + 6(1/16) + 1/16 = 8/16 = 1/2