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>. | ||
| − | ==Solution== | + | ==Video Solution (Mathematical Dexterity)== |
| + | https://www.youtube.com/watch?v=C14f91P2pYc | ||
==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}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 17:56, 18 February 2022
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 
. When either Azar or Carl plays either Jon or Sergey, Azar or Carl will win the match with probability 
. Assume that outcomes of different matches are independent. The probability that Carl will win the tournament is 
, where 
and 
are relatively prime positive integers. Find 
.
Video Solution (Mathematical Dexterity)
https://www.youtube.com/watch?v=C14f91P2pYc
See Also
| 2022 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
