Unit Circle Chord Probability: POTW #180 Sept. 7, 2015

[anemone]

Here is this week's POTW:

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Two points are picked at random on the unit circle $x^2+y^2=1$. What is the probability that the chord joining the two points has length at least 1?

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

Congratulations to the following members for their correct solution:):

1. MarkFL
2. lfdahl

Solution from MarkFL:

Spoiler

By symmetry, let us only consider the upper half of the circle. WLOG, let one of the points be at (1,0). We find the angle subtended at the origin of the point whose chord has a length of 1 to be $$\frac{\pi}{3}$$. Hence the probability of the chord having a length of at least 1 must be:

$$P(x)=\frac{2}{3}$$