2017 AMC 12A Problems/Problem 10
Problem
Chloé chooses a real number uniformly at random from the interval . Independently, Laurent chooses a real number uniformly at random from the interval
. What is the probability that Laurent's number is greater than Chloe's number?
Solution
Suppose Laurent's number is in the interval . Then, by symmetry, the probability of Laurent's number being greater is
. Next, suppose Laurent's number is in the interval
. 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
,
. 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.