fixed length random chord outside of circle.

Question

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.

Answer 1

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