2018 AIME II #13 (problem request)

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Misha rolls a standard, fair six-sided die until she rolls $1-2-3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Denote $P_i$ as the probability of rolling $1-2-3$ in $3$ consecutive rolls for an $n$ number of total rolls. Also denote $x = P_3+P_5+\cdots = 1-(P_4+P_6+\cdots)$. Note that\[P_{2n+1}=P_{2n}-\frac{P_{2n-2}}{6^3}\]for all $n \ge 2$. Summing this for $n=2,4,...$, we obtain\[x-\frac{1}{6^3} = \frac{6^3-1}{6^3}(1-x)\]\[\implies x = \frac{216}{431}\] Finally $m+n=216+431=\boxed{647}$. 



  1. Anonymous1/21/2022

    oh i requested this problem

    thanks for the solution :)

    you are very helpful


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