Difference between revisions of "2019 AIME I Problems/Problem 5"
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− | The | + | ==Problem 5== |
+ | A moving particle starts at the point <math>(4,4)</math> and moves until it hits one of the coordinate axes for the first time. When the particle is at the point <math>(a,b)</math>, it moves at random to one of the points <math>(a-1,b)</math>, <math>(a,b-1)</math>, or <math>(a-1,b-1)</math>, each with probability <math>\frac{1}{3}</math>, independently of its previous moves. The probability that it will hit the coordinate axes at <math>(0,0)</math> is <math>\frac{m}{3^n}</math>, where <math>m</math> and <math>n</math> are positive integers such that <math>m</math> is not divisible by <math>3</math>. Find <math>m + n</math>. | ||
+ | |||
+ | ==Solution 1== | ||
+ | One could recursively compute the probabilities of reaching <math>(0,0)</math> as the first axes point from any point <math>(x,y)</math> as <cmath>P(x,y) = \frac{1}{3} P(x-1,y) + \frac{1}{3} P(x,y-1) + \frac{1}{3} P(x-1,y-1)</cmath> for <math>x,y \geq 1,</math> and the base cases are | ||
+ | <math>P(0,0) = 1, P(x,0) = P(y,0) = 0</math> for any <math>x,y</math> not equal to zero. | ||
+ | We then recursively find <math>P(4,4) = \frac{245}{2187}</math> so the answer is <math>245 + 7 = \boxed{252}</math>. | ||
+ | |||
+ | |||
+ | |||
+ | If this algebra seems intimidating, you can watch a nice pictorial explanation of this by On The Spot Stem. | ||
+ | https://www.youtube.com/watch?v=XBRuy3_TM9w | ||
+ | |||
+ | ==Solution 2== | ||
+ | |||
+ | Obviously, the only way to reach (0,0) is to get to (1,1) and then have a <math>\frac{1}{3}</math> chance to get to (0,0). Let x denote a move left 1 unit, y denote a move down 1 unit, and z denote a move left and down one unit each. The possible cases for these moves are <math>(x,y,z)=(0,0,3),(1,1,2),(2,2,1)</math> and <math>(3,3,0)</math>. This gives a probability of <math>1 \cdot \frac{1}{27} + \frac{4!}{2!} \cdot \frac{1}{81} + \frac{5!}{2! \cdot 2!} \cdot \frac{1}{243} +\frac{6!}{3! \cdot 3!} \cdot \frac{1}{729}=\frac{245}{729}</math> to get to <math>(1,1)</math>. The probability of reaching <math>(0,0)</math> is <math>\frac{245}{3^7}</math>. This gives <math>245+7=\boxed{252}</math>. | ||
+ | |||
+ | ==Video Solution== | ||
+ | Unique solution: https://youtu.be/I-8xZGhoDUY | ||
+ | |||
+ | ~Shreyas S | ||
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− | |||
==See Also== | ==See Also== | ||
{{AIME box|year=2019|n=I|num-b=4|num-a=6}} | {{AIME box|year=2019|n=I|num-b=4|num-a=6}} | ||
+ | |||
+ | [[Category:Intermediate Probability Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 14:37, 16 October 2020
Problem 5
A moving particle starts at the point and moves until it hits one of the coordinate axes for the first time. When the particle is at the point , it moves at random to one of the points , , or , each with probability , independently of its previous moves. The probability that it will hit the coordinate axes at is , where and are positive integers such that is not divisible by . Find .
Solution 1
One could recursively compute the probabilities of reaching as the first axes point from any point as for and the base cases are for any not equal to zero. We then recursively find so the answer is .
If this algebra seems intimidating, you can watch a nice pictorial explanation of this by On The Spot Stem. https://www.youtube.com/watch?v=XBRuy3_TM9w
Solution 2
Obviously, the only way to reach (0,0) is to get to (1,1) and then have a chance to get to (0,0). Let x denote a move left 1 unit, y denote a move down 1 unit, and z denote a move left and down one unit each. The possible cases for these moves are and . This gives a probability of to get to . The probability of reaching is . This gives .
Video Solution
Unique solution: https://youtu.be/I-8xZGhoDUY
~Shreyas S
See Also
2019 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
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.