# Expected length of arc in a randomly divided circle

Choose, at random, three points on the unit circle. Interpret them as cuts that divide the circle into three arcs. Compute the expected length of the arc that contains the point (1, 0).

Generalise for N points.

• Not a proof, but an intuitive guess: the problem is equivalent to the total length of the first and last segment of a line of length $2\pi$ that has been randomly cut into four pieces. By symmetry each piece is expected to be $\frac{\pi}{2}$ long, for an answer of $\pi$ ($\frac{4\pi}{n+1}$ in the general case.) – user7530 Jan 8 '13 at 17:23
• Argue by symmetry. – André Nicolas Jan 8 '13 at 17:31

The length of the arc from $(1,0)$ to the next point is the minimum $M$ of three (or $n$) independent uniform random numbers $X_1,\ldots, X_n$ from $[0,2\pi)$. Since $P(M>x)=\prod P(X_i>x)=\left(1-\frac x{2\pi}\right)^n$, we find $$E(M) = \int_0^{2\pi}P(M>x)dx =\left. -\frac{2\pi}{n+1}\left(1-\frac x{2\pi}\right)^{n+1}\right|_0^{2\pi}=\frac{2\pi}{n+1}.$$ The same holds for the arc length to the the other arc end, hence the expected length of the full arc containing $(0,1)$ is $$\frac{4\pi}{n+1}.$$

• For 3 cuts, this reasoning gives you an expected length of $\pi$. However, with just 2 cuts, the expected length of one of the resulting 2 pieces chosen at random is $\pi$. For 3 cuts, it must be less. I don't think you can treat $(1,0)$ like a cut point that you then stitch back together. It really just signifies random selection of a segment after the fact. – Zimul8r Jan 8 '13 at 18:08
• That's not true. If the circle is cut into two pieces, one much bigger than the other, $(1,0)$ is more likely to be in the bigger piece. – user7530 Jan 8 '13 at 19:44
• Right. OK, so from math.stackexchange.com/questions/14190/… and math.stackexchange.com/questions/13959/…, in the 2 cut case, the pieces, being the longest and the shortest by default, have expected length $\frac{\pi}{2}$ and $\frac{3\pi}{2}$ with prob of containing $(1,0)$ at $25\%$ and $75\%$, resp. So the expected length would be $\left(\frac{1}{4}\right) \left(\frac{\pi}{2}\right) + \left(\frac{3}{4}\right) \left(\frac{3\pi}{2}\right) = \frac{5\pi}{4}$? – Zimul8r Jan 8 '13 at 20:16
• Shouldn't the random variables lie from (0, Pi) as after pi the length again start decreasing? – Rishabh Jain Sep 14 '18 at 18:43
• @RishabhJain To make it clear, I first look for the anti-clockwise next, not the overall closest cut point; therefore up to $2\pi$ is possible. – Hagen von Eitzen Sep 15 '18 at 8:55

You can cut the circle at (1,0) point and make it a line. The both side of the line corresponds to point (1,0) in the unit circle.

We also know that the answer for the circle is $$2\pi$$ times the answer of the uniform[0,1] case.

For uniform[0,1] case with the cutting method mentioned above, The expectation is $$E(min(X_1,...,X_n)+1-max(X_1,...,X_n))$$

We know that in this case $$E(min(X_1,...,X_n))=\dfrac{1}{n+1}$$ and $$E(max(X_1,...,X_n)) = \dfrac{n}{n+1}$$, so we have the final answer as

$$2\pi (1+\dfrac{1}{n+1}-\dfrac{n}{n+1})=\dfrac{4\pi}{n+1}$$