2017 AMC 10A Problem 15

Problem 15 of 25IntermediateCounting & Probability

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

If Laurent chooses a number in the interval

(2017, 4034],

then there is no way that Chloé can have the greater number. This means that Laurent has a

\frac{1}{2}

chance of automatically winning. Otherwise, Laurent chooses a number in the interval

[0, 2017].

The probability that he gets a greater number than Chloé is the same as Chloé getting a greater number than Laurent. This means that Laurent has a

\frac{1}{2}

chance of getting a greater number (when working with real intervals, the probability of a tie is essentially

0

due to the infinite size of the intervals). Laurent’s total chance of getting a greater number is

\dfrac{1}{2} \cdot 1 + \dfrac{1}{2} \cdot \dfrac{1}{2} = \dfrac{3}{4}.

Thus, C is the correct answer.