2017 AMC 12A Problem 10

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?

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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.