Rolling Right Along

Suppose we roll \(3\) fair, standard six-sided dice. Any dice that show a \(6\) are removed, after which the remaining dice, (if any), are rolled again. As before, any dice showing a \(6\) are removed, and any remaining dice are rolled again. We repeat this process until all \(3\) dice have been removed.

The expected number of rolls needed until all \(3\) dice have been removed is \(\dfrac{a}{b}\), where \(a\) and \(b\) are positive coprime integers. Find \(a - b.\)

Image Credit: Wikimedia Walter J. Pilsak
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