Rolling Kiriti's dice

We are given that two fair dice are thrown and the sum of the numbers appearing is greater than or equal to \(10\). If a third fair dice is then thrown, the probability that the sum of all three numbers appearing is greater than or equal to \(15\) can be expressed as \(\frac {a}{b}\) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a+b\)?

This problem is shared by Kiriti M. from the BdMO Divisional Olympiad.

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