Four standard, six faced dice are rolled. If the product of their values turns out to be an even number then the probability that their sum is odd, is :

If the product is even, hence at least one number must be even and since sum is odd. So, there can be 2 cases
Case I: 1 odd + 3even
Case II : 3 odd, 1 even
Total ways =  ${(}^4{C}_3{.3}^3.3=324)\times 2ways$  
probability = $\frac{648}{6^4-3^4}=\frac{648}{1215}=\frac{8}{15}$