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Difference between revisions of "2022 AIME II Problems/Problem 2"
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Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability <math>\frac23</math>. When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability <math>\frac34</math>. Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>.
 
Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability <math>\frac23</math>. When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability <math>\frac34</math>. Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>.
  
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==Solution 1==
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==Solution==
  
 
Let <math>A</math> be Azar, <math>C</math> be Carl, <math>J</math> be Jon, and <math>S</math> be Sergey. The <math>4</math> circles represent the <math>4</math> players, and the arrow is from the winner to the loser with the winning probability as the label.
 
Let <math>A</math> be Azar, <math>C</math> be Carl, <math>J</math> be Jon, and <math>S</math> be Sergey. The <math>4</math> circles represent the <math>4</math> players, and the arrow is from the winner to the loser with the winning probability as the label.
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This problem can be solved by using <math>2</math> cases.
 
This problem can be solved by using <math>2</math> cases.
  
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<math>\textbf{Case 1:}</math> <math>C</math>'s opponent for the semifinals is <math>A</math>
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<math>\textbf{Case 1:}</math> <math>C</math>'s opponent for the semifinal is <math>A</math>
  
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The probability <math>C</math>'s opponent is <math>A</math> is <math>\frac13</math>. Therefore the probability <math>C</math> wins the semifinals in this case is <math>\frac13 \cdot \frac13</math>. The other semifinal game is played between <math>J</math> and <math>S</math>, it doesn't matter who wins because <math>C</math> has the same probability of winning either one. The probability of <math>C</math> winning in the finals is <math>\frac34</math>, so the probability of <math>C</math> winning the tournament in case 1 is <math>\frac13 \cdot \frac13 \cdot \frac34</math>
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The probability <math>C</math>'s opponent is <math>A</math> is <math>\frac13</math>. Therefore the probability <math>C</math> wins the semifinal in this case is <math>\frac13 \cdot \frac13</math>. The other semifinal game is played between <math>J</math> and <math>S</math>, it doesn't matter who wins because <math>C</math> has the same probability of winning either one. The probability of <math>C</math> winning in the final is <math>\frac34</math>, so the probability of <math>C</math> winning the tournament in case 1 is <math>\frac13 \cdot \frac13 \cdot \frac34</math>
  
  
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<math>\textbf{Case 2:}</math> <math>C</math>'s opponent for the semifinals is <math>J</math> or <math>S</math>
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<math>\textbf{Case 2:}</math> <math>C</math>'s opponent for the semifinal is <math>J</math> or <math>S</math>
  
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It doesn't matter if <math>C</math>'s opponent is <math>J</math> or <math>S</math> because <math>C</math> has the same probability of winning either one. The probability <math>C</math>'s opponent is <math>J</math>/<math>S</math> is <math>\frac23</math>. Therefore the probability <math>C</math> wins the semifinals in this case is <math>\frac23 \cdot \frac34</math>. The other semifinal game is played between <math>A</math> and <math>J</math> or <math>S</math>. In this case it matters who wins in the other semifinal game because the probability of <math>C</math> winning <math>A</math> and <math>J</math> or <math>S</math> is different.  
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It doesn't matter if <math>C</math>'s opponent is <math>J</math> or <math>S</math> because <math>C</math> has the same probability of winning either one. The probability <math>C</math>'s opponent is <math>J</math> or <math>S</math> is <math>\frac23</math>. Therefore the probability <math>C</math> wins the semifinal in this case is <math>\frac23 \cdot \frac34</math>. The other semifinal game is played between <math>A</math> and <math>J</math> or <math>S</math>. In this case it matters who wins in the other semifinal game because the probability of <math>C</math> winning <math>A</math> and <math>J</math> or <math>S</math> is different.  
  
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<math>\textbf{Case 2.1:}</math> <math>C</math>'s opponent for the finals is <math>A</math>
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<math>\textbf{Case 2.1:}</math> <math>C</math>'s opponent for the final is <math>A</math>
  
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For this to happen, <math>A</math> must have won <math>J</math>/<math>S</math> in the semifinals, the probability is <math>\frac34</math>. Therefore, the probability that <math>C</math> won <math>A</math> in the finals is <math>\frac34 \cdot \frac13</math>.
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For this to happen, <math>A</math> must have won <math>J</math> or <math>S</math> in the semifinal, the probability is <math>\frac34</math>. Therefore, the probability that <math>C</math> won <math>A</math> in the final is <math>\frac34 \cdot \frac13</math>.
  
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<math>\textbf{Case 2.1:}</math> <math>C</math>'s opponent for the finals is <math>J</math> or <math>S</math>
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<math>\textbf{Case 2.2:}</math> <math>C</math>'s opponent for the final is <math>J</math> or <math>S</math>
  
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For this to happen, <math>J</math>/<math>S</math> must have won <math>A</math> in the semifinals, the probability is <math>\frac14</math>. Therefore, the probability that <math>C</math> won <math>J</math>/<math>S</math> in the finals is <math>\frac14 \cdot \frac34</math>.
+
For this to happen, <math>J</math> or <math>S</math> must have won <math>A</math> in the semifinal, the probability is <math>\frac14</math>. Therefore, the probability that <math>C</math> won <math>J</math> or <math>S</math> in the final is <math>\frac14 \cdot \frac34</math>.
  
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In Case 2 the probability of <math>C</math> winning the tournament is <math>\frac23 \cdot \frac3 \cdot (\frac34 \cdot \frac13 + \frac14 \cdot \frac34)</math>
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In Case 2 the probability of <math>C</math> winning the tournament is <math>\frac23 \cdot \frac34 \cdot (\frac34 \cdot \frac13 + \frac14 \cdot \frac34)</math>
  
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Adding case 1 and case 2 together we get <math>\frac13 \cdot \frac13 \cdot \frac34 + \frac23 \cdot \frac3 \cdot (\frac34 \cdot \frac13 + \frac14 \cdot \frac34) = \frac{29}{96}</math>. So the answer is <math>29 + 96 = \boxed{\textbf{125}}</math>
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Adding case 1 and case 2 together we get <math>\frac13 \cdot \frac13 \cdot \frac34 + \frac23 \cdot \frac34 \cdot (\frac34 \cdot \frac13 + \frac14 \cdot \frac34) = \frac{29}{96},</math> so the answer is <math>29 + 96 = \boxed{\textbf{125}}</math>.
  
 
~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
 
~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
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==Video Solution (Mathematical Dexterity)==
 
==Video Solution (Mathematical Dexterity)==
 
https://www.youtube.com/watch?v=C14f91P2pYc
 
https://www.youtube.com/watch?v=C14f91P2pYc
 +
 +
==Video Solution by Power of Logic==
 +
https://youtu.be/vvCes96cPm4
  
 
==See Also==
 
==See Also==
 
{{AIME box|year=2022|n=II|num-b=1|num-a=3}}
 
{{AIME box|year=2022|n=II|num-b=1|num-a=3}}
 +
 +
[[Category:Intermediate Combinatorics Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 15:15, 24 January 2023

Problem

Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probability $\frac23$. When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability $\frac34$. Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Solution

Let $A$ be Azar, $C$ be Carl, $J$ be Jon, and $S$ be Sergey. The $4$ circles represent the $4$ players, and the arrow is from the winner to the loser with the winning probability as the label.

2022AIMEIIP2.png

This problem can be solved by using $2$ cases.

$\textbf{Case 1:}$ $C$'s opponent for the semifinal is $A$

The probability $C$'s opponent is $A$ is $\frac13$. Therefore the probability $C$ wins the semifinal in this case is $\frac13 \cdot \frac13$. The other semifinal game is played between $J$ and $S$, it doesn't matter who wins because $C$ has the same probability of winning either one. The probability of $C$ winning in the final is $\frac34$, so the probability of $C$ winning the tournament in case 1 is $\frac13 \cdot \frac13 \cdot \frac34$


$\textbf{Case 2:}$ $C$'s opponent for the semifinal is $J$ or $S$

It doesn't matter if $C$'s opponent is $J$ or $S$ because $C$ has the same probability of winning either one. The probability $C$'s opponent is $J$ or $S$ is $\frac23$. Therefore the probability $C$ wins the semifinal in this case is $\frac23 \cdot \frac34$. The other semifinal game is played between $A$ and $J$ or $S$. In this case it matters who wins in the other semifinal game because the probability of $C$ winning $A$ and $J$ or $S$ is different.

$\textbf{Case 2.1:}$ $C$'s opponent for the final is $A$

For this to happen, $A$ must have won $J$ or $S$ in the semifinal, the probability is $\frac34$. Therefore, the probability that $C$ won $A$ in the final is $\frac34 \cdot \frac13$.

$\textbf{Case 2.2:}$ $C$'s opponent for the final is $J$ or $S$

For this to happen, $J$ or $S$ must have won $A$ in the semifinal, the probability is $\frac14$. Therefore, the probability that $C$ won $J$ or $S$ in the final is $\frac14 \cdot \frac34$.

In Case 2 the probability of $C$ winning the tournament is $\frac23 \cdot \frac34 \cdot (\frac34 \cdot \frac13 + \frac14 \cdot \frac34)$

Adding case 1 and case 2 together we get $\frac13 \cdot \frac13 \cdot \frac34 + \frac23 \cdot \frac34 \cdot (\frac34 \cdot \frac13 + \frac14 \cdot \frac34) = \frac{29}{96},$ so the answer is $29 + 96 = \boxed{\textbf{125}}$.

~isabelchen

Video Solution (Mathematical Dexterity)

https://www.youtube.com/watch?v=C14f91P2pYc

Video Solution by Power of Logic

https://youtu.be/vvCes96cPm4

See Also

2022 AIME II (Problems • Answer Key • Resources)
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Problem 1 		Followed by
Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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