2018 AIME II Problems/Problem 13

Problem

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 $m+n$ .

Solution

Call the probability you win on a certain toss $f_n$ , where $n$ is the toss number. Obviously, since the sequence has length 3, $f_1=0$ and $f_2=0$ . Additionally, $f_3=\left(\frac{1}{6}\right)^3$ . We can call this value $x$ , to keep our further equations looking clean. We can now write our general form for $f$ as $f_n=x(1-\sum_{i=1}^{n-3}f_i)$ . This factors the probability of the last 3 rolls being 1-2-3, and the important probability that the sequence has not been rolled in the past (because then the game would already be over). Note that $\sum_{i=1}^{\infty}f_i=1$ since you'll win at some point. An intermediate step here is figuring out $f_n-f_{n+1}$ . This is equal to $x(1-\sum_{i=1}^{n-3}f_i)-x(1-\sum_{i=1}^{n-2}f_i)=x(\sum_{i=1}^{n-2}f_i-\sum_{i=1}^{n-3}f_i)=xf_{n-2}$ . Adding up all the differences, i.e. $\sum_{i=2}^{\infty}(f_{2n-1}-f_{2n})$ will give us the amount by which the odds probability exceeds the even probability. Since they sum to 1, that means the odds probability will be half of the difference above one-half. Subbing in our earlier result from the intermediate step, the odd probability is equal to $\frac{1}{2}+\frac{1}{2}x\sum_{i=2}^{\infty}f_{2n-3}$ . Another way to find the odd probability is simply summing it up, which turns out to be $\sum_{i=1}^{\infty}f_{2n-1}$ . Note the infinite sums in both expressions are equal; let's call it $P$ . Equating them gives $\frac{1}{2}+\frac{1}{2}xP=P$ , or $P=\frac{1}{2-x}$ . Finally, substituting $x=\frac{1}{216}$ , we find that $P=\frac{216}{431}\approx 50.116\%$ . The sum of the numerator and denominator is thus $647$ and that is the answer. --DanDan0101

2018 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
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2018 AIME II Problems/Problem 13

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