A probability problem by Soumava Pal

Probability Level 3

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

AA gets the first chance to roll the die ,followed by BB, and then CC, followed by AA again if none of them gets a six.

Let the probability of AA winning the game be ab\dfrac{a}{b}, that of BB be cd\dfrac{c}{d} and that of CC winning the game be ef\dfrac{e}{f}, with a,b,c,d,e,fa,b,c,d,e,f being positive integers, with gcd(a,b)=gcd(c,d)=gcd(e,f)=1\gcd(a,b)=\gcd(c,d)=\gcd(e,f)=1.

Find a+b+c+d+e+fa+b+c+d+e+f.


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