Difference between revisions of "2017 AMC 12B Problems/Problem 20"
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Real numbers <math>x</math> and <math>y</math> are chosen independently and uniformly at random from the interval <math>(0,1)</math>. What is the probability that <math>\lfloor\log_2x\rfloor=\lfloor\log_2y\rfloor</math>? | Real numbers <math>x</math> and <math>y</math> are chosen independently and uniformly at random from the interval <math>(0,1)</math>. What is the probability that <math>\lfloor\log_2x\rfloor=\lfloor\log_2y\rfloor</math>? | ||
Revision as of 13:56, 15 February 2021
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
. 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
See Also
2017 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 19 |
Followed by Problem 21 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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