Dominating Dice

Probability Level 5

Two players each roll a pair of standard 4-sided dice. The rolls for the first player are $a_1, a_2$ with $a_1 \leq a_2$ and the rolls for the second player are $b_1,b_2$ with $b_1 \leq b_2$. The probability that $a_1 > b_1$ and $a_2 > b_2$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are coprime positive integers. What is the value of $a + b$?

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