2003 AMC 12B Problems/Problem 25
Problem
Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distance between the points are less than the radius of the circle?
Solution
The first point anywhere on the circle, because it doesn't matter where it is chosen.
The next point must lie within degrees of arc on either side, a total of
degrees possible, giving a total
chance. The last point must lie within
degrees of both.
The minimum area of freedom we have to place the third point is a degrees arc(if the first two are
degrees apart), with a
probability.
The maximum amount of freedom we have to place the third point is a
degree arc(if the first two are the same point), with a
probability.
As the second point moves farther away from the first point, up to a maximum of degrees, the probability changes linearly (every degree it moves, adds one degree to where the third could be).
Therefore, we can average probabilities at each end to find to find the average probability we can place the third point based on a varying second point.
Therefore the total probability is or
See Also
2003 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
Followed by Last Problem |
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