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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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Solution There must be a reason for the D3 rating. | Comment 2 of 10 |
If the independent randomization of the positions of the points is done using a uniform distribution along the circumference of the circle, then the probability is 1/3, as 1/3 of the points selected for B are withing 60° of arc of point A.

However, if the randomization procedure is not uniform, the probability will be higher, as point A will be more likely than not to be in an area of concentrated probability, making it more likely that point B will be closer than would be the case in a uniform distribution.

Of course it would be a psychological problem to find the probability distribution of choices of randomization procedures, so no overall probability can be given other than it's at least 1/3.

  Posted by Charlie on 2018-04-26 10:16:46
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