Difference between revisions of "2017 AMC 12A Problems/Problem 10"
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==Problem== | ==Problem== | ||
| − | + | 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)}\ \dfrac{1}{2} \qquad\textbf{(B)}\ \dfrac{2}{3} \qquad\textbf{(C)}\ \dfrac{3}{4} \qquad\textbf{(D)}\ \dfrac{5}{6} \qquad\textbf{(E)}\ \dfrac{7}{8} </math> | <math> \textbf{(A)}\ \dfrac{1}{2} \qquad\textbf{(B)}\ \dfrac{2}{3} \qquad\textbf{(C)}\ \dfrac{3}{4} \qquad\textbf{(D)}\ \dfrac{5}{6} \qquad\textbf{(E)}\ \dfrac{7}{8} </math> | ||
Revision as of 21:56, 14 September 2018
Problem
Chloe chooses a real number uniformly at random from the interval ![$[ 0,2017 ]$](http://latex.artofproblemsolving.com/c/2/d/c2de3743d64138d4bd7f371fb3de265718621d07.png)
. Independently, Laurent chooses a real number uniformly at random from the interval ![$[ 0 , 4034 ]$](http://latex.artofproblemsolving.com/a/9/1/a917fb72d07cd62fd999921ebed24aebddad250d.png)
. What is the probability that Laurent's number is greater than Chloe's number?
Solution
Suppose Laurent's number is in the interval ![$[ 0, 2017 ]$](http://latex.artofproblemsolving.com/7/6/a/76a259806417a270b1859d1dd2db61fb0090d670.png)
. Then, by symmetry, the probability of Laurent's number being greater is 
. Next, suppose Laurent's number is in the interval ![$[ 2017, 4034 ]$](http://latex.artofproblemsolving.com/2/2/e/22eb67bd191cf44619936c4b6abe568adb550602.png)
. Then Laurent's number will be greater with probability 
. Since each case is equally likely, the probability of Laurent's number being greater is 
, so the answer is C.
Alternate Solution: Geometric Probability
Let 
be the number chosen randomly by Chloé. Because it is given that the number Chloé chooses is in the interval , 
. Next, let 
be the number chosen randomly by Laurent. Because it is given that the number Laurent chooses is in the interval ![$[ 0, 4034 ]$](http://latex.artofproblemsolving.com/2/d/a/2da5b6a2bc5003d0fb9c7e7b4f5b52e9291b50b0.png)
, 
. Since we are looking for when Laurent's number is greater than Chloé's we write the equation 
. When these three inequalities are graphed the area captured by and represents all the possibilities, forming a rectangle 2017 in width and 4034 in height. Thus making its area 
. The area captured by , , and represents the possibilities of Laurent winning, forming a trapezoid with a height 2017 in length and bases 4034 and 2017 length, thus making an area 
. The simplified quotient of these two areas is the probability Laurent's number is larger than Chloé's, which is 
.
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 | ||
| 2017 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 9 |
Followed by Problem 11 |
| 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 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
