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Difference between revisions of "2017 AMC 10A Problems/Problem 15"

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==Problem==
 
==Problem==
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Chloé chooses a real number uniformly at random from the interval <math>[0, 2017]</math>. Independently, Laurent chooses a real number uniformly at random from the interval <math>[0, 4034]</math>. What is the probability that Laurent's number is greater than Chloé's number? (Assume they cannot be equal)
 
  
−
<math> \mathrm{\textbf{(A)} \ }\frac{1}{2}\qquad \mathrm{\textbf{(B)} \ } \frac{2}{3}\qquad \mathrm{\textbf{(C)} \ } \frac{3}{4}\qquad \mathrm{\textbf{(D)} \ } \frac{5}{6}\qquad \mathrm{\textbf{(E)} \ }\frac{7}{8}</math>
+
Chloe chooses a real number uniformly at random from the interval <math>[0, 2017]</math>. Independently, Laurent chooses a real number uniformly at random from the interval <math>[0, 4034]</math>. What is the probability that Laurent's number is greater than Chloe's number?
 +
 
 +
<math> \textbf{(A) } \frac{1}{2} \qquad \textbf{(B) } \frac{2}{3} \qquad \textbf{(C) } \frac{3}{4} \qquad \textbf{(D) } \frac{5}{6} \qquad \textbf{(E) } \frac{7}{8}</math>
  
 
==Solution 1==
 
==Solution 1==
 
Denote "winning" to mean "picking a greater number".
 
Denote "winning" to mean "picking a greater number".
−
There is a <math>\frac{1}{2}</math> chance that Laurent chooses a number in the interval <math>[2018, 4034]</math>. In this case, Chloé cannot possibly win, since the maximum number she can pick is <math>2017</math>. Otherwise, if Laurent picks a number in the interval <math>[0, 2017]</math>, with probability <math>\frac{1}{2}</math>, then the two people are symmetric, and each has a <math>\frac{1}{2}</math> chance of winning. Then, the total probability is <math>\frac{1}{2}*1 + \frac{1}{2}*\frac{1}{2} = \boxed{\textbf{(C)}\ \frac{3}{4}}</math>
+
There is a <math>\frac{1}{2}</math> chance that Laurent chooses a number in the interval <math>[2017, 4034]</math>. In this case, Chloé cannot possibly win, since the maximum number she can pick is <math>2017</math>. Otherwise, if Laurent picks a number in the interval <math>[0, 2017]</math>, with probability <math>\frac{1}{2}</math>, then the two people are symmetric, and each has a <math>\frac{1}{2}</math> chance of winning. Then, the total probability is: <math>\frac{1}{2}\times1 + \frac{1}{2}\times\frac{1}{2} = \boxed{\textbf{(C)}\ \frac{3}{4}}.</math>
 +
 
 +
~Small grammar mistake corrected by virjoy2001 (missing period), small error corrected by Terribleteeth, small grammar error corrected by Astro2010~
  
 
==Solution 2==
 
==Solution 2==
 
We can use geometric probability to solve this.
 
We can use geometric probability to solve this.
−
Suppose a point <math>(x,y)</math> lies in the <math>xy</math>-plane. Let <math>x</math> be Chloe's number and <math>y</math> be Laurent's number. Then obviously we want <math>y>x</math>, which basically gives us a region above a line. We know that Chloe's number is in the interval <math>[0,2017]</math> and Laurent's number is in the interval <math>[0,4034]</math>, so we can create a rectangle in the plane, whose length is <math>2017</math> and whose width is <math>4034</math>. Drawing it out, we see that it is easier to find the probability that Chloe's number is greater than Laurent's number and subtract this probability from <math>1</math>. The probability that Chloe's number is larger than Laurent's number is simply the area of the region under the line <math>y>x</math>, which is <math>\frac{2017 \cdot 2017}{2}</math>. Instead of bashing this out we know that the rectangle has area <math>2017 \cdot 4034</math>. So the probability that Laurent has a smaller number is <math>\frac{2017 \cdot 2017}{2 \cdot 2017 \cdot 4034}</math>. Simplifying the expression yields <math>\frac{1}{4}</math> and so <math>1-\frac{1}{4}= \boxed{\textbf{(C)}\ \frac{3}{4}}</math>.
+
Suppose a point <math>(x,y)</math> lies in the <math>xy</math>-plane. Let <math>x</math> be Chloe's number and <math>y</math> be Laurent's number. Then obviously we want <math>y>x</math>, which basically gives us a region above a line. We know that Chloe's number is in the interval <math>[0,2017]</math> and Laurent's number is in the interval <math>[0,4034]</math>, so we can create a rectangle in the plane, whose length is <math>2017</math> and whose width is <math>4034</math>. Drawing it out and dividing into 4 congruent triangles, we see that Laurent's winning area is 3 triangles and Chloe's is 1 triangle. <math>\boxed{\textbf{(C)}\ \frac{3}{4}}</math>.
  
 
==Solution 3==
 
==Solution 3==
−
Scale down by <math>2017</math> to get that Chloe picks from <math>[0,1]</math> and Laurent picks from <math>[0,2]</math>. There are an infinite number of cases for the number that Chloe picks, but they are all centered around the average of <math>0.5</math>. Therefore, Laurent has a range of <math>0.5</math> to <math>2</math> to pick from, on average, which is a length of <math>2-0.5=1.5</math> out of a total length of <math>2-0=2</math>. Therefore, the probability is <math>1.5/2=15/20=\boxed{3/4 \space \text{(C)}}</math>
+
Scale down by <math>2017</math> to get that Chloe picks from <math>[0,1]</math> and Laurent picks from <math>[0,2]</math>. There are an infinite number of cases for the number that Chloe picks, but they are all centered around the average of <math>0.5</math>. Therefore, Laurent has a winning range of <math>[X, 2]</math>, where the average value of <math>X</math> is <math>0.5</math>. Thus the average winning length is <math>2-0.5=1.5</math> out of a total length of <math>2-0=2</math>. Therefore, the probability is <math>1.5/2=15/20=\boxed{\frac{3}{4} \space \text{(C)}}.</math>
 +
 
 +
==Solution 4==
 +
In total there are <math>2018\times4034</math> ways in which Laurene and Chloe can choose numbers (as same number cannot be chosen by both). If Chloe chooses 2017, then Lauren has 2017 ways to win, if Chloe chooses 2016, Lauren has 2018 ways to win and so on until if Chloe chooses 0, Lauren has 4034 ways to win. Thus the answer is:
 +
 
 +
<math>\frac{\text{number of ways lauren can win}}{\text{number of combinations}}=\frac{2017+2018+2019...+4034}{2018\times4034}</math>
 +
 
 +
Using arithmetic series formula:
 +
 
 +
<math>\frac{\frac{1}{2}(2018)(2017+4034)}{2018\times4034}=\frac{3}{4}</math>
 +
 
 +
<math>\fbox{C}</math> is the correct answer.
 +
 
 +
~ Lion08
 +
 
 +
NOTE: Because the problem says real numbers, not integers, this solution may not always work.
 +
 
 +
==Video Solution==
 +
A video solution for this can be found here: https://www.youtube.com/watch?v=PQFNwW1XFaQ
 +
 
 +
https://youtu.be/NB4KXQiqgi0
 +
 
 +
~savannahsolver
 +
 
 +
==Video Solution 2==
 +
https://www.youtube.com/watch?v=s4vnGlwwHHw&t=840s
 +
 
 +
== Video Solution ==
 +
https://youtu.be/IRyWOZQMTV8?t=4163
 +
 
 +
~ pi_is_3.14
  
 
==See Also==
 
==See Also==

Latest revision as of 13:30, 13 October 2024

Problem

Chloe chooses a real number uniformly at random from the interval $[0, 2017]$. Independently, Laurent chooses a real number uniformly at random from the interval $[0, 4034]$. What is the probability that Laurent's number is greater than Chloe's number?

$\textbf{(A) } \frac{1}{2} \qquad \textbf{(B) } \frac{2}{3} \qquad \textbf{(C) } \frac{3}{4} \qquad \textbf{(D) } \frac{5}{6} \qquad \textbf{(E) } \frac{7}{8}$

Solution 1

Denote "winning" to mean "picking a greater number". There is a $\frac{1}{2}$ chance that Laurent chooses a number in the interval $[2017, 4034]$. In this case, Chloé cannot possibly win, since the maximum number she can pick is $2017$. Otherwise, if Laurent picks a number in the interval $[0, 2017]$, with probability $\frac{1}{2}$, then the two people are symmetric, and each has a $\frac{1}{2}$ chance of winning. Then, the total probability is: $\frac{1}{2}\times1 + \frac{1}{2}\times\frac{1}{2} = \boxed{\textbf{(C)}\ \frac{3}{4}}.$

~Small grammar mistake corrected by virjoy2001 (missing period), small error corrected by Terribleteeth, small grammar error corrected by Astro2010~

Solution 2

We can use geometric probability to solve this. Suppose a point $(x,y)$ lies in the $xy$-plane. Let $x$ be Chloe's number and $y$ be Laurent's number. Then obviously we want $y>x$, which basically gives us a region above a line. We know that Chloe's number is in the interval $[0,2017]$ and Laurent's number is in the interval $[0,4034]$, so we can create a rectangle in the plane, whose length is $2017$ and whose width is $4034$. Drawing it out and dividing into 4 congruent triangles, we see that Laurent's winning area is 3 triangles and Chloe's is 1 triangle. $\boxed{\textbf{(C)}\ \frac{3}{4}}$.

Solution 3

Scale down by $2017$ to get that Chloe picks from $[0,1]$ and Laurent picks from $[0,2]$. There are an infinite number of cases for the number that Chloe picks, but they are all centered around the average of $0.5$. Therefore, Laurent has a winning range of $[X, 2]$, where the average value of $X$ is $0.5$. Thus the average winning length is $2-0.5=1.5$ out of a total length of $2-0=2$. Therefore, the probability is $1.5/2=15/20=\boxed{\frac{3}{4} \space \text{(C)}}.$

Solution 4

In total there are $2018\times4034$ ways in which Laurene and Chloe can choose numbers (as same number cannot be chosen by both). If Chloe chooses 2017, then Lauren has 2017 ways to win, if Chloe chooses 2016, Lauren has 2018 ways to win and so on until if Chloe chooses 0, Lauren has 4034 ways to win. Thus the answer is:

$\frac{\text{number of ways lauren can win}}{\text{number of combinations}}=\frac{2017+2018+2019...+4034}{2018\times4034}$

Using arithmetic series formula:

$\frac{\frac{1}{2}(2018)(2017+4034)}{2018\times4034}=\frac{3}{4}$

$\fbox{C}$ is the correct answer.

~ Lion08

NOTE: Because the problem says real numbers, not integers, this solution may not always work.

Video Solution

A video solution for this can be found here: https://www.youtube.com/watch?v=PQFNwW1XFaQ

https://youtu.be/NB4KXQiqgi0

~savannahsolver

Video Solution 2

https://www.youtube.com/watch?v=s4vnGlwwHHw&t=840s

Video Solution

https://youtu.be/IRyWOZQMTV8?t=4163

~ pi_is_3.14

See Also

2017 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Problem 16
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

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