Four points in a circle

Geometry Level pending

Four points, A, B, C, and D are chosen uniformly and at random in a unit circle.

What is the probability that D is closer to A than it is to either of the other two points?

Please provide your answer to 3 decimal places.

The answer is 0.333.

1 solution

Geoff Pilling
May 21, 2017

Consider that A, B, and C are chosen uniformly and at random, and now you need to choose D.

By symmetry, it is equiprobable that it is closest to A, B, or C. So, the probability of it being closest to each one is $\frac{1}{3} \approx \boxed{0.333}$

With the Geometry label I was looking for some kind of catch, but couldn't find one so went with $1/3$ . (I wouldn't have hesitated if it was labeled as a Discrete Mathematics question, so good choice of label. :))

Potential follow-up question: With two points A and B chosen uniformly and independently at random in a circle, what is the probability that B is closer to A than it is to the nearest point on the circle's circumference? If A were fixed at the origin then the probability would be $1/4$ , but when A moves from the center the points that are equidistant from A and the circumference form a sort of distorted oval shape. I suspect there might be a clever approach to this one....

Brian Charlesworth - 4 years ago

Ah yes, a much more thought provoking question! ;-) (And, yes, I wasn't really sure which category to put this in)

Geoff Pilling - 4 years ago

Hmmm... I wonder exactly what is the locus of points that are the same distance from a unit circle and, say, the point $(0,\frac{1}{2})$

Geoff Pilling - 4 years ago

This is tricky. I've created a parameter $\theta$ which is the angle a diameter makes with the $x$ -axis. The locus is then the set of points

$(x,y) = \left(\dfrac{3\cos(\theta)}{8 - 4\sin(\theta)}, \dfrac{3\sin(\theta)}{8 - 4\sin(\theta)}\right)$ .

Then $x^{2} + y^{2} = \dfrac{9}{(8 - 4\sin(\theta))^{2}}$ . I'd like to eliminate the parameter but I can't see how. It sort of looks like an ellipse.

Brian Charlesworth - 4 years ago

Four points in a circle

The answer is 0.333.

1 solution

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