Difference between revisions of "2014 AMC 12B Problems/Problem 22"

m (same person as poster, made some of the parentheses bigger, and a few minor edits)
m (I apologize for the many posts, I'm new to Latex so I'm just trying to format correctly.)
Line 6: Line 6:
  
 
We therefore seek to rewrite P(1) in terms of P(5), using the fact that
 
We therefore seek to rewrite P(1) in terms of P(5), using the fact that
 +
  
 
P(N) = <math>\frac {N} {10}P(N - 1) + \frac {10 - N} {N}P(N + 1)</math>
 
P(N) = <math>\frac {N} {10}P(N - 1) + \frac {10 - N} {N}P(N + 1)</math>
 +
  
 
as said in the problem.
 
as said in the problem.
 +
  
 
Hence P(1) = <math>\frac {1} {10}P(0) + \frac {9} {10}P(2) = \frac {9} {10}P(2)</math>
 
Hence P(1) = <math>\frac {1} {10}P(0) + \frac {9} {10}P(2) = \frac {9} {10}P(2)</math>
 +
  
 
<math>\Rightarrow P(2) = \frac {10} {9}P(1)</math>
 
<math>\Rightarrow P(2) = \frac {10} {9}P(1)</math>
 +
  
 
Returning to our original equation:
 
Returning to our original equation:
 +
  
 
<math>P(1) = \frac {9} {10}P(2) = \frac {9} {10}\left(\frac{2} {10}P(1) + \frac{8} {10}P(3)\right)</math>
 
<math>P(1) = \frac {9} {10}P(2) = \frac {9} {10}\left(\frac{2} {10}P(1) + \frac{8} {10}P(3)\right)</math>
 +
  
 
<math>= \frac {9} {50}P(1) + \frac {18} {25}P(3) \Rightarrow P(1) - \frac {9} {50}P(1)</math>  
 
<math>= \frac {9} {50}P(1) + \frac {18} {25}P(3) \Rightarrow P(1) - \frac {9} {50}P(1)</math>  
 
<math>= \frac {18} {25}P(3)</math>
 
<math>= \frac {18} {25}P(3)</math>
 +
  
 
<math>\Rightarrow P(3) = \frac {41} {36}P(1)</math>
 
<math>\Rightarrow P(3) = \frac {41} {36}P(1)</math>
 +
  
 
Returning to our original equation:
 
Returning to our original equation:
 +
  
 
<math>P(1) = \frac {9} {50}P(1) + \frac {18} {25}\left(\frac {3} {10}P(2) + \frac {7} {10}P(4)\right)</math>
 
<math>P(1) = \frac {9} {50}P(1) + \frac {18} {25}\left(\frac {3} {10}P(2) + \frac {7} {10}P(4)\right)</math>
 +
  
 
<math>= \frac {9} {50}P(1) + \frac {27} {125}P(2) + \frac {63} {125}P(4)</math>
 
<math>= \frac {9} {50}P(1) + \frac {27} {125}P(2) + \frac {63} {125}P(4)</math>
 +
  
 
<math>= \frac {9} {50}P(1) + \frac {27} {125}\left(\frac {10} {9}P(1)\right) + \frac {63} {125}\left(\frac {4} {10}P(3) + \frac {6} {10}P(5)\right)</math>
 
<math>= \frac {9} {50}P(1) + \frac {27} {125}\left(\frac {10} {9}P(1)\right) + \frac {63} {125}\left(\frac {4} {10}P(3) + \frac {6} {10}P(5)\right)</math>
 +
  
 
Cleaing up the coefficients, we have:  
 
Cleaing up the coefficients, we have:  
 +
  
 
<math>= \frac {21} {50}P(1) + \frac {126} {625}P(3) + \frac {189} {625}P(5)</math>
 
<math>= \frac {21} {50}P(1) + \frac {126} {625}P(3) + \frac {189} {625}P(5)</math>
  
<math>= \frac {21} {50}P(1) + \frac {126} {625}(\frac {41} {36}P(1)) + \frac {189} {625}(\frac {1} {2})</math>
+
 
 +
<math>= \frac {21} {50}P(1) + \frac {126} {625}\left(\frac {41} {36}P(1)\right) + \frac {189} {625}\left(\frac {1} {2}\right)</math>
 +
 
  
 
Hence, P(1) = <math>\frac {525} {1250}P(1) + \frac {287} {1250}P(1) + \frac {189} {1250}</math>
 
Hence, P(1) = <math>\frac {525} {1250}P(1) + \frac {287} {1250}P(1) + \frac {189} {1250}</math>
 +
  
 
<math>\Rightarrow P(1) - \frac {812} {1250}P(1) = \frac {189} {1250} \Rightarrow P(1) = \frac {189} {438}</math>
 
<math>\Rightarrow P(1) - \frac {812} {1250}P(1) = \frac {189} {1250} \Rightarrow P(1) = \frac {189} {438}</math>
 +
 +
  
 
= <math>\boxed{\frac {63} {146}\, (C)}</math>
 
= <math>\boxed{\frac {63} {146}\, (C)}</math>
 +
 +
 +
-Dukejukem

Revision as of 20:50, 20 February 2014

A long, but straightforward bash:

Define P(N) to be the probability that the frog survives starting from pad N.

Then note that by symmetry, P(5) = 1/2, since the probabilities of the frog moving subsequently in either direction from pad 5 are equal.

We therefore seek to rewrite P(1) in terms of P(5), using the fact that


P(N) = $\frac {N} {10}P(N - 1) + \frac {10 - N} {N}P(N + 1)$


as said in the problem.


Hence P(1) = $\frac {1} {10}P(0) + \frac {9} {10}P(2) = \frac {9} {10}P(2)$


$\Rightarrow P(2) = \frac {10} {9}P(1)$


Returning to our original equation:


$P(1) = \frac {9} {10}P(2) = \frac {9} {10}\left(\frac{2} {10}P(1) + \frac{8} {10}P(3)\right)$


$= \frac {9} {50}P(1) + \frac {18} {25}P(3) \Rightarrow P(1) - \frac {9} {50}P(1)$ $= \frac {18} {25}P(3)$


$\Rightarrow P(3) = \frac {41} {36}P(1)$


Returning to our original equation:


$P(1) = \frac {9} {50}P(1) + \frac {18} {25}\left(\frac {3} {10}P(2) + \frac {7} {10}P(4)\right)$


$= \frac {9} {50}P(1) + \frac {27} {125}P(2) + \frac {63} {125}P(4)$


$= \frac {9} {50}P(1) + \frac {27} {125}\left(\frac {10} {9}P(1)\right) + \frac {63} {125}\left(\frac {4} {10}P(3) + \frac {6} {10}P(5)\right)$


Cleaing up the coefficients, we have:


$= \frac {21} {50}P(1) + \frac {126} {625}P(3) + \frac {189} {625}P(5)$


$= \frac {21} {50}P(1) + \frac {126} {625}\left(\frac {41} {36}P(1)\right) + \frac {189} {625}\left(\frac {1} {2}\right)$


Hence, P(1) = $\frac {525} {1250}P(1) + \frac {287} {1250}P(1) + \frac {189} {1250}$


$\Rightarrow P(1) - \frac {812} {1250}P(1) = \frac {189} {1250} \Rightarrow P(1) = \frac {189} {438}$


= $\boxed{\frac {63} {146}\, (C)}$


-Dukejukem