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 ?
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