UNSW MathSoc Championships Q14

You have three dice each with numbers $1-6$ . The first die is a fair die and the second die has $1/12$ probability of showing each of $1$ , $3$ and $6$ , and $1/4$ probability of showing each of $2$ , $4$ and $5$ . The third die has probability $7/25$ of showing $1$ , probability $1/5$ of showing each of $2$ and $6$ , probability $3/25$ of showing each of $3$ and $5$ , and probability $2/25$ of showing $4$ . You roll the three dice. Assuming the results are independent, what is the probability that the sum of the results is even?

If your answer is of the form $\frac{a}{b}$ , where $a$ and $b$ are positive, coprime integers, find the value of $a+b$ .