A discrete mathematics problem by Soumava Pal

Discrete Mathematics Level 3

\(A\), \(B\) and \(C\) take turns rolling a fair die (with equal probability of it showing any of \(\{1,2,3,4,5,6\}\), and the as soon as one of them gets a six, the game ends and he wins.

\(A\) gets the first chance to roll the die ,followed by \(B\), and then \(C\), followed by \(A\) again if none of them gets a six.

Let the probability of \(A\) winning the game be \(\dfrac{a}{b}\), that of \(B\) be \(\dfrac{c}{d}\) and that of \(C\) winning the game be \(\dfrac{e}{f}\), with \(a,b,c,d,e,f\) being positive integers, with \(\gcd(a,b)=\gcd(c,d)=\gcd(e,f)=1\).

Find \(a+b+c+d+e+f\).


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