Question

Chloe 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 Chloe's number?

$\dfrac{1}{2}$

$\dfrac{2}{3}$

$\dfrac{3}{4}$

$\dfrac{5}{6}$

$\dfrac{7}{8}$

Solution:

If Laurent chooses a number in the interval $(2017, 4034],$ then there is no way that Chloe 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 she gets a greater number than Chloe is the same as Chloe getting a greater number then 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.

2017 AMC 10A Problem 15

Solution:

Problem 15 in Other Years