2017 AMC 12B Problems/Problem 20
Problem
Real numbers and are chosen independently and uniformly at random from the interval . What is the probability that ?
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 . Recall that the probability that or is the case, where case and case are mutually exclusive, is the sum of each individual probability. Symbolically that's . Thus, the probability we are looking for is the sum of the probability for each of the cases . 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 .
Solution by: vedadehhc \\ Edited by: jingwei325
See Also
2017 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 19 |
Followed by Problem 21 |
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