Difference between revisions of "2016 AIME I Problems/Problem 13"
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| − | + | ==Problem== | |
| + | Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line <math>y = 24</math>. A fence is located at the horizontal line <math>y = 0</math>. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point where <math>y=0</math>, with equal likelihoods he chooses one of three directions where he either jumps parallel to the fence or jumps away from the fence, but he never chooses the direction that would have him cross over the fence to where <math>y < 0</math>. Freddy starts his search at the point <math>(0, 21)</math> and will stop once he reaches a point on the river. Find the expected number of jumps it will take Freddy to reach the river. | ||
==Solution== | ==Solution== | ||
Notice that we don't really care about what the <math>x</math>-coordinate of the frog is. So let's let <math>f(y)</math> denote the expected number of times Freddy will jump at a <math>y</math> coordinate of <math>y</math> until he reaches the line <math>y = 24</math>. So therefore we want to find <math>f(21)</math>. | Notice that we don't really care about what the <math>x</math>-coordinate of the frog is. So let's let <math>f(y)</math> denote the expected number of times Freddy will jump at a <math>y</math> coordinate of <math>y</math> until he reaches the line <math>y = 24</math>. So therefore we want to find <math>f(21)</math>. | ||
Revision as of 16:03, 4 March 2016
Problem
Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line 
. A fence is located at the horizontal line 
. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point where 
, with equal likelihoods he chooses one of three directions where he either jumps parallel to the fence or jumps away from the fence, but he never chooses the direction that would have him cross over the fence to where 
. Freddy starts his search at the point 
and will stop once he reaches a point on the river. Find the expected number of jumps it will take Freddy to reach the river.
Solution
Notice that we don't really care about what the 
-coordinate of the frog is. So let's let 
denote the expected number of times Freddy will jump at a 
coordinate of until he reaches the line . So therefore we want to find 
.
So we have 
. Suppose Freddy is at 
. He has a 
probability of moving horizontally, 
chance of moving up and chance of moving down. So we have
![\[f(n) = 1 + \frac{1}{2} f(n) + \frac{1}{4} f(n-1) + \frac{1}{4} f(n+1)\]](http://latex.artofproblemsolving.com/0/7/b/07bdc090f5d83bb0e38799bb9cf9e2caac280d6f.png)
So we get the recursion 
. Rearranging we see 
. That means the difference between consecutive terms goes down by 
each time. So for convenience let's let 
and 
. So that means
![\[f(k) = z + d + (d-4) + (d-8) + \cdots + (d - 4(k-1)) = z + kd - 2k(k-1)\]](http://latex.artofproblemsolving.com/a/5/2/a528be0725763d7af38d3ab8a40185285685452b.png)
Yes, this is a quadratic. Now, notice that since there is a boundary, we have to give special care to 
. We have 
so this turns into 
and this results in 
. So now we know
![\[f(k) = z - k - 2k^2\]](http://latex.artofproblemsolving.com/4/1/6/416abbf57cf6674997e91869cd73e42b20804077.png)
Now, we also have that so that gives us 
so 
. So now we know 
so plugging in 
we get 
See also
| 2016 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 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. 
