# 2017 AMC 12B Problems/Problem 20

## Problem 20

Real numbers and are chosen independently and uniformly at random from the interval . What is the probability that , where denotes the greatest integer less than or equal to the real number ?

## Solution

First let us take the case that . In this case, both and lie in the interval . The probability of this is . Similarly, in the case that , and lie in the interval , and the probability is . It is easy to see that the probabilities for for are the infinite geometric series that starts at and with common ratio . Using the formula for the sum of an infinite geometric series, we get that the probability is .