Difference between revisions of "2024 SSMO Relay Round 5 Problems/Problem 2"
(Created page with "==Problem== Let <math>T = TNYWR.</math> In the game of high and low, the computer chooses two integers without replacement from the set <math>\{1,2,3,\dots,T\}</math>. The co...") |
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| + | Note that if the computer chooses <math>k\leq6,</math> then there is a <math>\frac{12-k}{11}</math> chance that the player guesses correctly and if <math>k\geq 6,</math> then there is a <math>\frac{k-1}{11}</math> chance that the player guesses correctly. Thus, the answer is <cmath>\frac{2(6+\cdots+11)}{11\cdot12} = \frac{17}{22}\implies 17+22 = \boxed{39}.</cmath> | ||
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| + | ~SMO_Team | ||
Latest revision as of 14:48, 10 September 2025
Problem
Let 
In the game of high and low, the computer chooses two integers without replacement from the set 
. The computer displays the first integer and asks the player if the second integer is higher or lower. Given that the player always plays optimally, the chances of guessing correctly is 
for relatively prime positive integers 
and 
Find 
Solution
Note that if the computer chooses 
then there is a 
chance that the player guesses correctly and if 
then there is a 
chance that the player guesses correctly. Thus, the answer is ![\[\frac{2(6+\cdots+11)}{11\cdot12} = \frac{17}{22}\implies 17+22 = \boxed{39}.\]](http://latex.artofproblemsolving.com/d/b/f/dbf5c3f969a6742ca8a9c45e1681472ad82f61e2.png)
~SMO_Team
