Difference between revisions of "2007 iTest Problems/Problem 59"
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== Problem == | == Problem == | ||
| − | Let <math>T=\text{TNFTPP}</math>. Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers such that <math>a/b<1/2</math>. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play <math>\textit{Probabicloneme}</math> so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is <math>(T- | + | Let <math>T=\text{TNFTPP}</math>. Fermi and Feynman play the game <math>\textit{Probabicloneme}</math> in which Fermi wins with probability <math>a/b</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers such that <math>a/b<1/2</math>. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play <math>\textit{Probabicloneme}</math> so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is <math>(T-322)/(2T-601)</math>. Find the value of <math>a</math>. |
== Solution == | == Solution == | ||
Revision as of 08:12, 16 April 2024
Problem
Let 
. Fermi and Feynman play the game 
in which Fermi wins with probability 
, where 
and 
are relatively prime positive integers such that 
. The rest of the time Feynman wins (there are no ties or incomplete games). It takes a negligible amount of time for the two geniuses to play so they play many many times. Assuming they can play infinitely many games (eh, they're in Physicist Heaven, we can bend the rules), the probability that they are ever tied in total wins after they start (they have the same positive win totals) is 
. Find the value of .
Solution
17
See Also
| 2007 iTest (Problems, Answer Key) | ||
| Preceded by: Problem 58 |
Followed by: Problem 60 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • TB1 • TB2 • TB3 • TB4 | ||
