Difference between revisions of "2017 AMC 12B Problems/Problem 20"
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<math>\lfloor \\rfloor = -1</math><math>1/2<<1</math>. that <math>\lfloor \\rfloor = \lfloor \\rfloor = -</math><math>/</math> <math>\\2</math><math>141</math>. that <math>\lfloor \\rfloor = \lfloor \\rfloor = -</math> <math>14</math> common ratio <math>14</math>. is <math>141 - 14= D13</math>
Revision as of 21:55, 16 February 2017
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 ?
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 .