The sample space for the problem is all the outcomes where the sum of the two dice is 8; the "good" outcomes are the ones that have a 6 on one of the two dice.
6+2
5+3
4+4
3+5
2+6
There are 5 ways to get a sum of 8; 2 of those ways have 6 on one of the two dice.
The probability that one of the dice shows 6, given that the sum of the two dice is 8, is 2/5.
Using the formal mathematical definition of conditional probability....
P(one of the dice is 6 | the sum of the two dice is 8)
= P((one of the dice is 6) AND (the sum of the dice is 8)) 2/36 2
-------------------------------------------------------- = ------ = ---
P(sum of the dice is 8) 5/36 5