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

(Solution 3)
(Solution 3)
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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:
 
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:
  
(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.
+
(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 <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>  

Revision as of 14:22, 12 February 2021

Problem 5

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

Video Solution

Unique solution: https://youtu.be/I-8xZGhoDUY

~Shreyas S

See Also

2019 AIME I (ProblemsAnswer KeyResources)
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

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