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consider a uniform distribution on a unit circle, I construct a cord by the following steps:

  1. pick one endpoint A within the unit circle uniformly.
  2. points that are $0<d<1$ distance away from (assume Euclidean) A forms another circle. Pick another endpoint B uniformly on the circle. The length of the cord is $d$.

Now, the chord might not lie within the circle (B is outside of unit circle), I am wondering what is the probability of this bad event, and the case for higher dimension.

I think this question can be approached by integrating: $$ P( \text{B outside of unit circle}) = \int_{\text{circle}} P( B \ outside|position\ of\ A) P(position\ of\ A) $$

However, I need some help with what the first term in the integral should be.

Thank you in advance.

Gilles Bonnet
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fast tooth
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  • Isn't the "cord" with its length $d$ actually irrelevant? Wouldn't this be equivalent to: "I draw a unit circle B with center randomly selected inside another circle A, and I pick a random point in B, what is the probability that it's outside A"? – leonbloy Apr 07 '14 at 19:02

1 Answers1

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Let $t$ be the distance between the centers. Then $p_t(t) = 2 t$ ($0\le t \le 1$). Now, to compute the probability that the second point is outside the first circle, we need to compute the area of intersection, which is $2 \cos^{-1}(t/2)$

Then $$P = \int_0^1 \frac{1- 2\cos^{-1}(t/2)}{\pi} \ 2 t \, dt$$

leonbloy
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  • thanks, you revise the question more intuitively and give the correct answer, thanks so much. In case of higher dimension, I find the volume of intersection in the post:http://math.stackexchange.com/questions/162250/how-to-compute-the-volume-of-intersection-between-two-hyperspheres – fast tooth Apr 07 '14 at 19:43
  • @johnniac Thanks. I fixed an error with $p_t(t)$ - there might be more. – leonbloy Apr 07 '14 at 19:45