Difference between revisions of "2018 AIME II Problems/Problem 13"
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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 <math>\dfrac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | 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 <math>\dfrac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
− | ==Solution== | + | ==Solution 1== |
Call the probability you win on a certain toss <math>f_n</math>, where <math>n</math> is the toss number. | Call the probability you win on a certain toss <math>f_n</math>, where <math>n</math> is the toss number. | ||
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Finally, substituting <math>x=\frac{1}{216}</math>, we find that <math>P=\frac{216}{431}</math>, giving us a final answer of <math>216 + 431 = \boxed{647}</math>. | Finally, substituting <math>x=\frac{1}{216}</math>, we find that <math>P=\frac{216}{431}</math>, giving us a final answer of <math>216 + 431 = \boxed{647}</math>. | ||
--DanDan0101 | --DanDan0101 | ||
+ | |||
+ | ==Solution 2 (Recursion)== | ||
+ | Let <math>S(n)</math> be the number of strings of length <math>n</math> containing the digits <math>1</math> through <math>6</math> that do not contain the string <math>123</math>. Then we have <math>S(n) = 6 \cdot S(n-1) - S(n-3)</math> because we can add any digit to end of a string with length <math>n-1</math> but we have to subtract all the strings that end in <math>123</math>. We rewrite this as | ||
+ | \begin{align*} | ||
+ | S(n) &= 6 \cdot S(n-1) - S(n-3) \\ &= 6 \cdot (6 \cdot S(n-2) - S(n-4)) - (6 \cdot (S(n-4) - S(n-6)) \\ &= 36 \cdot S(n-2) - 12 \cdot S(n-4) + S(n-6) | ||
+ | \end{align*}wish to compute <math>P=\sum_{n=0}^\infty \frac{S(2n)}{6^{2n+3}}</math> since the last three rolls are <math>123</math> for the game to end. Summing over the recursion, we obtain | ||
+ | <cmath>\sum_{n=0}^\infty \frac{S(2n)}{6^{2n+3}} =36 \cdot \sum_{n=0}^\infty \frac{S(2n-2)}{6^{2n+3}} - 12 \cdot \sum_{n=0}^\infty \frac{S(2n-4)}{6^{2n+3}}+ \sum_{n=0}^\infty \frac{S(2n-6)}{6^{2n+3}} </cmath>Now shift the indices so that the inside term is the same: | ||
+ | \begin{align*} | ||
+ | \sum_{n=3}^\infty \frac{S(2n)}{6^{2n+3}} &= \frac{36}{6^2} \cdot \sum_{n=2}^\infty \frac{S(2n)}{6^{2n+3}} - \frac{12}{6^4} \cdot \sum_{n=1}^\infty \frac{S(2n)}{6^{2n+3}} + \frac{1}{6^6} \cdot \sum_{n=0}^\infty \frac{S(2n)}{6^{2n+3}} \\ \left(P - \frac{S(0)}{6^3} - \frac{S(2)}{6^5} -\frac{S(4)}{6^7} \right) &= \frac{36}{6^2} \cdot \left( P - \frac{S(0)}{6^3} - \frac{S(2)}{6^5}\right) - \frac{12}{6^4} \cdot \left( P - \frac{S(0)}{6^3} \right) + \frac{1}{6^6} \cdot P | ||
+ | \end{align*}Note that <math>S(0) = 1, S(2) = 26</math> and <math>S(4) = 6^4 - 2 \cdot 6 = 1284</math>. Therefore, | ||
+ | \begin{align*} | ||
+ | \left(P - \frac{1}{6^3} - \frac{36}{6^5} -\frac{1284}{6^7} \right) = \frac{36}{6^2} \cdot \left( P - \frac{1}{6^3} - \frac{36}{6^5}\right) - \frac{12}{6^4} \cdot \left( P - \frac{1}{6^3} \right) + \frac{1}{6^6} \cdot P | ||
+ | \end{align*}Solving for <math>P</math>, we obtain <math>P = \frac{216}{431} \implies m+n = \boxed{647}</math>. -Vfire | ||
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{{AIME box|year=2018|n=II|num-b=12|num-a=14}} | {{AIME box|year=2018|n=II|num-b=12|num-a=14}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 08:29, 26 March 2018
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 where and are relatively prime positive integers. Find .
Solution 1
Call the probability you win on a certain toss , where is the toss number. Obviously, since the sequence has length 3, and . Additionally, . We can call this value , to keep our further equations looking clean. We can now write our general form for as . 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 since you'll win at some point. An intermediate step here is figuring out . This is equal to . Adding up all the differences, i.e. 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 . Another way to find the odd probability is simply summing it up, which turns out to be . Note the infinite sums in both expressions are equal; let's call it . Equating them gives , or . Finally, substituting , we find that , giving us a final answer of . --DanDan0101
Solution 2 (Recursion)
Let be the number of strings of length containing the digits through that do not contain the string . Then we have because we can add any digit to end of a string with length but we have to subtract all the strings that end in . We rewrite this as \begin{align*} S(n) &= 6 \cdot S(n-1) - S(n-3) \\ &= 6 \cdot (6 \cdot S(n-2) - S(n-4)) - (6 \cdot (S(n-4) - S(n-6)) \\ &= 36 \cdot S(n-2) - 12 \cdot S(n-4) + S(n-6) \end{align*}wish to compute since the last three rolls are for the game to end. Summing over the recursion, we obtain Now shift the indices so that the inside term is the same: \begin{align*} \sum_{n=3}^\infty \frac{S(2n)}{6^{2n+3}} &= \frac{36}{6^2} \cdot \sum_{n=2}^\infty \frac{S(2n)}{6^{2n+3}} - \frac{12}{6^4} \cdot \sum_{n=1}^\infty \frac{S(2n)}{6^{2n+3}} + \frac{1}{6^6} \cdot \sum_{n=0}^\infty \frac{S(2n)}{6^{2n+3}} \\ \left(P - \frac{S(0)}{6^3} - \frac{S(2)}{6^5} -\frac{S(4)}{6^7} \right) &= \frac{36}{6^2} \cdot \left( P - \frac{S(0)}{6^3} - \frac{S(2)}{6^5}\right) - \frac{12}{6^4} \cdot \left( P - \frac{S(0)}{6^3} \right) + \frac{1}{6^6} \cdot P \end{align*}Note that and . Therefore, \begin{align*} \left(P - \frac{1}{6^3} - \frac{36}{6^5} -\frac{1284}{6^7} \right) = \frac{36}{6^2} \cdot \left( P - \frac{1}{6^3} - \frac{36}{6^5}\right) - \frac{12}{6^4} \cdot \left( P - \frac{1}{6^3} \right) + \frac{1}{6^6} \cdot P \end{align*}Solving for , we obtain . -Vfire
2018 AIME II (Problems • Answer Key • Resources) | ||
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