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

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{{duplicate|[[2014 AMC 12B Problems|2014 AMC 12B #22]] and [[2014 AMC 10B Problems|2014 AMC 10B #25]]}}
 
{{duplicate|[[2014 AMC 12B Problems|2014 AMC 12B #22]] and [[2014 AMC 10B Problems|2014 AMC 10B #25]]}}
  
==Problem==
+
== Problem ==
 
In a small pond there are eleven lily pads in a row labeled 0 through 10.  A frog is sitting on pad 1.  When the frog is on pad <math>N</math>, <math>0<N<10</math>, it will jump to pad <math>N-1</math> with probability <math>\frac{N}{10}</math> and to pad <math>N+1</math> with probability <math>1-\frac{N}{10}</math>.  Each jump is independent of the previous jumps.  If the frog reaches pad 0 it will be eaten by a patiently waiting snake.  If the frog reaches pad 10 it will exit the pond, never to return.  What is the probability that the frog will escape without being eaten by the snake?
 
In a small pond there are eleven lily pads in a row labeled 0 through 10.  A frog is sitting on pad 1.  When the frog is on pad <math>N</math>, <math>0<N<10</math>, it will jump to pad <math>N-1</math> with probability <math>\frac{N}{10}</math> and to pad <math>N+1</math> with probability <math>1-\frac{N}{10}</math>.  Each jump is independent of the previous jumps.  If the frog reaches pad 0 it will be eaten by a patiently waiting snake.  If the frog reaches pad 10 it will exit the pond, never to return.  What is the probability that the frog will escape without being eaten by the snake?
  
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\textbf{(E) }\frac{1}{2}\qquad</math>
 
\textbf{(E) }\frac{1}{2}\qquad</math>
  
 
+
== Solutions ==
==Solution 1==
+
=== Solution 1 ===
 
 
 
A long, but straightforward bash:
 
A long, but straightforward bash:
  
 
Define <math>P(N)</math> to be the probability that the frog survives starting from pad N.
 
Define <math>P(N)</math> to be the probability that the frog survives starting from pad N.
 
  
 
Then note that by symmetry, <math>P(5) = 1/2</math>, since the probabilities of the frog moving subsequently in either direction from pad 5 are equal.
 
Then note that by symmetry, <math>P(5) = 1/2</math>, since the probabilities of the frog moving subsequently in either direction from pad 5 are equal.
 
  
 
We therefore seek to rewrite <math>P(1)</math> in terms of <math>P(5)</math>, using the fact that
 
We therefore seek to rewrite <math>P(1)</math> in terms of <math>P(5)</math>, using the fact that
 
  
 
<math>P(N) = \frac {N} {10}P(N - 1) + \frac {10 - N} {10}P(N + 1)</math>
 
<math>P(N) = \frac {N} {10}P(N - 1) + \frac {10 - N} {10}P(N + 1)</math>
 
  
 
as said in the problem.
 
as said in the problem.
 
  
 
Hence <math>P(1) = \frac {1} {10}P(0) + \frac {9} {10}P(2) = \frac {9} {10}P(2)</math>
 
Hence <math>P(1) = \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}\left(\frac {41} {36}P(1)\right) + \frac {189} {625}\left(\frac {1} {2}\right)</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, <math>P(1) = \frac {525} {1250}P(1) + \frac {287} {1250}P(1) + \frac {189} {1250}</math>
 
Hence, <math>P(1) = \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>
 
 
  
 
Or set <math>P(1)=a,P(2)=b,P(3)=c,P(4)=d,P(5)=e=1/2</math>:
 
Or set <math>P(1)=a,P(2)=b,P(3)=c,P(4)=d,P(5)=e=1/2</math>:
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Since <math>e=\frac{1}{2}</math>, <math>\frac{73a}{63}=\frac{1}{2}\implies a=\boxed{\textbf{(C) }\frac{63}{146}}</math>.
 
Since <math>e=\frac{1}{2}</math>, <math>\frac{73a}{63}=\frac{1}{2}\implies a=\boxed{\textbf{(C) }\frac{63}{146}}</math>.
  
==Solution 2==
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=== Solution 2 ===
 
 
 
Notice that the probabilities are symmetrical around the fifth lily pad. If the frog is on the fifth lily pad, there is a <math>\frac{1}{2}</math> chance that it escapes and a <math>\frac{1}{2}</math> that it gets eaten. Now, let <math>P_k</math> represent the probability that the frog escapes if it is currently on pad <math>k</math>. We get the following system of <math>5</math> equations:
 
Notice that the probabilities are symmetrical around the fifth lily pad. If the frog is on the fifth lily pad, there is a <math>\frac{1}{2}</math> chance that it escapes and a <math>\frac{1}{2}</math> that it gets eaten. Now, let <math>P_k</math> represent the probability that the frog escapes if it is currently on pad <math>k</math>. We get the following system of <math>5</math> equations:
 
<cmath>P_1=\frac{9}{10}\cdot P_2</cmath>
 
<cmath>P_1=\frac{9}{10}\cdot P_2</cmath>
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<cmath>P_5=\frac{5}{10}</cmath>
 
<cmath>P_5=\frac{5}{10}</cmath>
 
We want to find <math>P_1</math>, since the frog starts at pad <math>1</math>. Solving the above system yields <math>P_1=\frac{63}{146}</math>, so the answer is <math>\boxed{(C)}</math>.
 
We want to find <math>P_1</math>, since the frog starts at pad <math>1</math>. Solving the above system yields <math>P_1=\frac{63}{146}</math>, so the answer is <math>\boxed{(C)}</math>.
==Video Solution==
+
 
 +
== Video Solution ==
 
https://www.youtube.com/watch?v=DMdgh2mMiWM
 
https://www.youtube.com/watch?v=DMdgh2mMiWM
 +
 
== See also ==
 
== See also ==
 
{{AMC10 box|year=2014|ab=B|num-b=24|after=Last Question}}
 
{{AMC10 box|year=2014|ab=B|num-b=24|after=Last Question}}
 
{{AMC12 box|year=2014|ab=B|num-b=21|num-a=23}}
 
{{AMC12 box|year=2014|ab=B|num-b=21|num-a=23}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 13:17, 19 January 2021

The following problem is from both the 2014 AMC 12B #22 and 2014 AMC 10B #25, so both problems redirect to this page.

Problem

In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake?

$\textbf{(A) }\frac{32}{79}\qquad \textbf{(B) }\frac{161}{384}\qquad \textbf{(C) }\frac{63}{146}\qquad \textbf{(D) }\frac{7}{16}\qquad \textbf{(E) }\frac{1}{2}\qquad$

Solutions

Solution 1

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

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

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

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

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

Or set $P(1)=a,P(2)=b,P(3)=c,P(4)=d,P(5)=e=1/2$: \[a=0.1\emptyset+0.9b,b=0.2a+0.8c,c=0.3b+0.7d,d=0.4c+0.6e\] \[10a=\emptyset+9b,10b=2a+8c,10c=3b+7d,10d=4c+6e\] \[\implies b=\frac{10a-\emptyset}{9},c=\frac{5b-a}{4},d=\frac{10c-3b}{7},e=\frac{5d-2c}{3}=1/2\] $b=\frac{10a}{9}$

$c=\frac{5\left(\frac{10a}{9}\right)-a}{4}=\frac{\frac{50a}{9}-a}{9}=\frac{41a}{36}$

$d=\frac{10\left(\frac{41a}{36}\right)-3\left(\frac{30a}{9}\right)}{7}=\frac{\frac{205a}{18}-\frac{10a}{3}}{7}=\frac{145a}{126}$

$e=\frac{5\left(\frac{145a}{126}\right)-2\left(\frac{41a}{36}\right)}{3}=\frac{\frac{725a}{126}-\frac{41a}{18}}{3}=\frac{73a}{63}$

Since $e=\frac{1}{2}$, $\frac{73a}{63}=\frac{1}{2}\implies a=\boxed{\textbf{(C) }\frac{63}{146}}$.

Solution 2

Notice that the probabilities are symmetrical around the fifth lily pad. If the frog is on the fifth lily pad, there is a $\frac{1}{2}$ chance that it escapes and a $\frac{1}{2}$ that it gets eaten. Now, let $P_k$ represent the probability that the frog escapes if it is currently on pad $k$. We get the following system of $5$ equations: \[P_1=\frac{9}{10}\cdot P_2\] \[P_2=\frac{2}{10}\cdot P_1 + \frac{8}{10}\cdot P_3\] \[P_3=\frac{3}{10}\cdot P_2 + \frac{7}{10}\cdot P_4\] \[P_4=\frac{4}{10}\cdot P_3 + \frac{6}{10}\cdot P_5\] \[P_5=\frac{5}{10}\] We want to find $P_1$, since the frog starts at pad $1$. Solving the above system yields $P_1=\frac{63}{146}$, so the answer is $\boxed{(C)}$.

Video Solution

https://www.youtube.com/watch?v=DMdgh2mMiWM

See also

2014 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 24
Followed by
Last Question
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
All AMC 10 Problems and Solutions
2014 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 21
Followed by
Problem 23
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
All AMC 12 Problems and Solutions

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