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Two random points (Posted on 2018-04-26) Difficulty: 3 of 5
On a circumference of a given circle with a radius R two random points A & B are independently chosen.

What is the probability of AB being less than R ?

No Solution Yet Submitted by Ady TZIDON    
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re(3): There must be a reason for the D3 rating. | Comment 6 of 10 |
(In reply to re(2): There must be a reason for the D3 rating. by Ady TZIDON)

Your mention of a probability of 1/4 struck me.  I couldn't (and still can't) imagine the probability would be less than 1/3, as any concentration of points in particular area(s) would only increase the probability of proximity.

You mentioned Bertrand's paradox in relation to this. The example that came up was with choosing a random chord--not the same thing as independently choosing two random points.  Here's the description that leads to a probability of 1/3:

The "random midpoint" method: Choose a point anywhere within the circle and construct a chord with the chosen point as its midpoint. The chord is longer than a side of the inscribed triangle if the chosen point falls within a concentric circle of radius 1/2 the radius of the larger circle. The area of the smaller circle is one fourth the area of the larger circle, therefore the probability a random chord is longer than a side of the inscribed triangle is 1/4.

However, using the endpoints of this chord as the two points means that the  two points have not been chosen independently; it is the chord that is chosen randomly, with the endpoints of that chord chosen as the two points--they are not independent.

  Posted by Charlie on 2018-04-26 12:04:02
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