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kdavis004
 Senior Boarder
Posts: 63 
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Mark off the points on the circumgerence of a circle clockwise from 0 to 1. For example, the top is 0; “east” is .25, the bottom is .5, and so forth.
You generate two random numbers between 0 and 1, mark them on the circle and draw a lime between them to form a chord.
You repeat this process with two other random numbers.
What is the probability that the two chords intersect?
Bonus question: What would be the average distance, as a fraction of the radius, between the center of the circle and the center of a chord drawn in this fashion?
Bill Ryan
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dagger29
 Senior Boarder
Posts: 78
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Spoiler follows.
I make it 1/3.
wlog the first point is at 0 and the second is <= 0.5.
So we need to evaluate
p=0.5 /
2 * 2*p*(1-p).dp
/ p=0
which comes to 1/3.
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glundby
Expert Boarder
Posts: 81
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. . . . . . . . . . . . . . . Say the first pair of points is (a0,a1) and the second pair is (b0,b1). If we write down the order of the points clockwise starting from 0, there are 24 possibilities. Of these, the ones where the chords intersect have the form a_i b_j a_(1-i) b_(1-j) (there are 4 of these), and similarly with the roles of a and b reversed. So it's 8/24 = 1/3.
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quest_marsman
Senior Boarder
Posts: 72
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Others have answered the first question; here's my attempt at a
Bonus spoiler:
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If I'm not mistaken, the average center-to-chord distance is 2/pi.
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Johnders
Expert Boarder
Posts: 88
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Original baffler retained as spoiler space.
Given any set of four points on a circle, we can label them A,B,C, and D in clockwise order. The other endpoint of the chord beginning at A can be either B, C, or D with equal probability, but the chords intersect iff it is C. So the probability is 1/3.
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Johnders
Expert Boarder
Posts: 88
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1/SQRT(2)?
We can limit our attention to one quadrant of the circle. Measuring in radians from 0 to pi/2, the mean would be pi/4. So the length of the line from the center of the circle to the center of the chord would be cos (pi/4) = 1/SQRT(2).
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Roger1955
Senior Boarder
Posts: 60
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Who said they had to intersect INSIDE the circle ?
Phil Brady
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imported_Adrian
Senior Boarder
Posts: 72
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A chord is a line segment inside a circle. The chords do not exist outside the circle. Therefore if the chords intersect they must do so inside the circle.
regards
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