2
$\begingroup$

If random points are chosen from a circle, what is the probability that all of them come from same semicircle ?

My reasoning is that; Let two random points be chosen first. Obviously then both of them lie in same semicircle. Now mark any diameter including those two on the same side of it. Any more point chosen will either lie within or beyond the semicircle made by diameter each with probability 1/2.

So 3 points lying on same semicircle is $ \frac{1}{2} $.

4points lying in same semicircle is $\frac{1}{2}.\frac{1}{2} = \frac{1}{2^2} $
Similarly, for n such points the total probability is $\frac{1}{2^{n-2}}$.

Am I right in my reasoning?

$\endgroup$
3
$\begingroup$

No, your reasoning is not quite right. The first two points lie in many semicircles - in fact an infinite number. What proportion of all semi-circles do they lie in? That depends on how close they are to each other.

This question is answered here

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.