Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2019 AIME I Problems/Problem 5"

(→‎Solution 3)
 
(6 intermediate revisions by 3 users not shown)
Line 1: Line 1:
−
==Problem 5==
+
==Problem==
 
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>.
 
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>.
  
Line 18: Line 18:
 
==Solution 3==
 
==Solution 3==
  
−
Since the particle stops at one of the axes, we know that the particle most pass through <math>(1,1)</math>. Thus, it suffices to consider the probability our particle will reach <math>(1,1)</math>. Denote a move to the left, down, diagonally, as X, Y, Z, respectively. Then the only ways to get to <math>(1,1)</math> from <math>(4,4)</math> are the following:
+
Since the particle stops at one of the axes, we know that the particle most pass through <math>(1,1)</math>. Thus, it suffices to consider the probability our particle will reach <math>(1,1)</math>. Then the only ways to get to <math>(1,1)</math> from <math>(4,4)</math> are the following:
  
−
<math>(1) 0X 0Y 3Z</math>
+
(1) 3 moves diagonally   
−
<math>(2) 1X 1Y 2Z</math>
+
 
−
<math>(3) 2X 2Y 1Z</math>
+
(2) 2 moves diagonally, 1 move left, 1 move down
−
<math>(4) 3X 3Y 0Z</math>
+
 
 +
(3) 1 move diagonally, 2 moves left and 2 moves down.
 +
 
 +
(4) 3 moves left, 3 moves down.
  
 
The probability of (1) is <math>\frac{1}{3^3}</math>. The probability of (2) is <math>\frac{\frac{4!}{2!}}{3^4} = \frac{12}{3^4}</math>. The probability of (3) is <math>\frac{\frac{5!}{2!2!}}{3^5} = \frac{30}{3^5}</math>. The probability of (4) is <math>\frac{\frac{6!}{3!3!}}{3^6} = \frac{20}{3^6}</math>. Adding all of these together, we obtain a total probability of <math>\frac{245}{3^6}</math> that our particle will hit <math>(1,1)</math>. Trivially, there is a <math>\frac{1}{3}</math> chance our particle will hit <math>(0,0)</math> from <math>(1,1)</math>. So our final probability will be <math>\frac{245}{3^6} \cdot \frac{1}{3} = \frac{245}{3^7} \implies m = 245, n = 7 \implies \boxed{252}</math>  
 
The probability of (1) is <math>\frac{1}{3^3}</math>. The probability of (2) is <math>\frac{\frac{4!}{2!}}{3^4} = \frac{12}{3^4}</math>. The probability of (3) is <math>\frac{\frac{5!}{2!2!}}{3^5} = \frac{30}{3^5}</math>. The probability of (4) is <math>\frac{\frac{6!}{3!3!}}{3^6} = \frac{20}{3^6}</math>. Adding all of these together, we obtain a total probability of <math>\frac{245}{3^6}</math> that our particle will hit <math>(1,1)</math>. Trivially, there is a <math>\frac{1}{3}</math> chance our particle will hit <math>(0,0)</math> from <math>(1,1)</math>. So our final probability will be <math>\frac{245}{3^6} \cdot \frac{1}{3} = \frac{245}{3^7} \implies m = 245, n = 7 \implies \boxed{252}</math>  
  
 
~NotSoTrivial
 
~NotSoTrivial
 +
==Solution 4 (Official MAA)==
 +
All paths that first hit the axes at the origin must pass through the point <math>(1,1)</math>. There are <math>63</math> paths from the point <math>(4,4)</math> to the point <math>(1,1)</math>: <math>\tbinom{6}{3}=20</math> that take <math>3</math> steps left and <math>3</math> steps down, <math>\tbinom{5}{2\,2\,1}=30</math> that take <math>2</math> steps left, <math>2</math> steps down, and <math>1</math> diagonal step, <math>\tbinom{4}{1\,1\,2}=12</math> steps that take <math>1</math> step left, <math>1</math> steps down, and <math>2</math> diagonal steps, and <math>1</math> that takes <math>3</math> diagonal steps. The total probability of moving from <math>(4,4)</math> to <math>(1,1)</math> is therefore <cmath>\frac{1}{3^6}\cdot20+\frac{1}{3^5}\cdot30+\frac{1}{3^4}\cdot12+\frac{1}{3^3}\cdot1=\frac{245}{3^6}.</cmath> Multiplying by <math>\tfrac13</math> gives <math>\tfrac{245}{3^7},</math> the probability that the path first reaches the axes at the origin. The requested sum is <math>245+7=252</math>.
 +
==Video Solution #1(A nice visual explanation)==
 +
https://youtu.be/JQdad7APQG8?t=1340
  
 
==Video Solution==
 
==Video Solution==

Latest revision as of 14:51, 25 February 2021

Problem

A moving particle starts at the point $(4,4)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $(a,b)$, it moves at random to one of the points $(a-1,b)$, $(a,b-1)$, or $(a-1,b-1)$, each with probability $\frac{1}{3}$, independently of its previous moves. The probability that it will hit the coordinate axes at $(0,0)$ is $\frac{m}{3^n}$, where $m$ and $n$ are positive integers such that $m$ is not divisible by $3$. Find $m + n$.

Solution 1

One could recursively compute the probabilities of reaching $(0,0)$ as the first axes point from any point $(x,y)$ as \[P(x,y) = \frac{1}{3} P(x-1,y) + \frac{1}{3} P(x,y-1) + \frac{1}{3} P(x-1,y-1)\] for $x,y \geq 1,$ and the base cases are $P(0,0) = 1, P(x,0) = P(y,0) = 0$ for any $x,y$ not equal to zero. We then recursively find $P(4,4) = \frac{245}{2187}$ so the answer is $245 + 7 = \boxed{252}$.


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 $\frac{1}{3}$ 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 $(x,y,z)=(0,0,3),(1,1,2),(2,2,1)$ and $(3,3,0)$. This gives a probability of $1 \cdot \frac{1}{27} + \frac{4!}{2!} \cdot \frac{1}{81} + \frac{5!}{2! \cdot 2!} \cdot \frac{1}{243} +\frac{6!}{3! \cdot 3!} \cdot \frac{1}{729}=\frac{245}{729}$ to get to $(1,1)$. The probability of reaching $(0,0)$ is $\frac{245}{3^7}$. This gives $245+7=\boxed{252}$.

Solution 3

Since the particle stops at one of the axes, we know that the particle most pass through $(1,1)$. Thus, it suffices to consider the probability our particle will reach $(1,1)$. Then the only ways to get to $(1,1)$ from $(4,4)$ are the following:

(1) 3 moves diagonally

(2) 2 moves diagonally, 1 move left, 1 move down

(3) 1 move diagonally, 2 moves left and 2 moves down.

(4) 3 moves left, 3 moves down.

The probability of (1) is $\frac{1}{3^3}$. The probability of (2) is $\frac{\frac{4!}{2!}}{3^4} = \frac{12}{3^4}$. The probability of (3) is $\frac{\frac{5!}{2!2!}}{3^5} = \frac{30}{3^5}$. The probability of (4) is $\frac{\frac{6!}{3!3!}}{3^6} = \frac{20}{3^6}$. Adding all of these together, we obtain a total probability of $\frac{245}{3^6}$ that our particle will hit $(1,1)$. Trivially, there is a $\frac{1}{3}$ chance our particle will hit $(0,0)$ from $(1,1)$. So our final probability will be $\frac{245}{3^6} \cdot \frac{1}{3} = \frac{245}{3^7} \implies m = 245, n = 7 \implies \boxed{252}$

~NotSoTrivial

Solution 4 (Official MAA)

All paths that first hit the axes at the origin must pass through the point $(1,1)$. There are $63$ paths from the point $(4,4)$ to the point $(1,1)$: $\tbinom{6}{3}=20$ that take $3$ steps left and $3$ steps down, $\tbinom{5}{2\,2\,1}=30$ that take $2$ steps left, $2$ steps down, and $1$ diagonal step, $\tbinom{4}{1\,1\,2}=12$ steps that take $1$ step left, $1$ steps down, and $2$ diagonal steps, and $1$ that takes $3$ diagonal steps. The total probability of moving from $(4,4)$ to $(1,1)$ is therefore \[\frac{1}{3^6}\cdot20+\frac{1}{3^5}\cdot30+\frac{1}{3^4}\cdot12+\frac{1}{3^3}\cdot1=\frac{245}{3^6}.\] Multiplying by $\tfrac13$ gives $\tfrac{245}{3^7},$ the probability that the path first reaches the axes at the origin. The requested sum is $245+7=252$.

Video Solution #1(A nice visual explanation)

https://youtu.be/JQdad7APQG8?t=1340

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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2019_AIME_I_Problems/Problem_5&oldid=147942"