Winning on a Crooked Roll

Two six-sided dice each have the numbers 1 through 6 on their faces. Neither die is fair, but they are both weighted the same. The probability of rolling a certain number on one die is given in the table below:

\(\begin{array}{c|cccccc} \mbox{number} & 1 & 2 & 3 & 4 & 5 & 6\\ \hline \mbox{probability} & \frac{1}{6} & \frac{1}{6} & \frac{1}{9} & ? & \frac{2}{9} & ?\\ \end{array}\)

If the probability that the two dice both show the same numbers is \(\left(\frac{2}{3}\right)^4\), we can express the probability of rolling 10 on these two dice as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b\)?

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