Question

Chloé 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 Chloé's number?

$\dfrac{1}{2}$

$\dfrac{2}{3}$

$\dfrac{3}{4}$

$\dfrac{5}{6}$

$\dfrac{7}{8}$

Solution:

With probability $\dfrac{1}{2},$ Laurent's number lies in $[2017,4034],$ which exceeds any number Chloé could choose, so he wins for certain.

With the other probability $\dfrac{1}{2},$ Laurent's number lies in $[0,2017],$ matching Chloé's interval; by symmetry he is larger half the time. The total probability is $\dfrac{1}{2}\cdot1+\dfrac{1}{2}\cdot\dfrac{1}{2}=\dfrac{3}{4}.$

Thus, the correct answer is C.

2017 AMC 12A Problem 10

Solution:

Problem 10 in Other Years