Problems

1. The point \(A\) lies on the circumference of a circle of radius \(a\) and centre \(O\). The point \(B\) is chosen at random on the circumference, so that the angle \(AOB\) has a uniform distribution on \([0, 2\pi]\). Find the expected length of the chord \(AB\).
2. The point \(C\) is chosen at random in the interior of a circle of radius \(a\) and centre \(O\), so that the probability that it lies in any given region is proportional to the area of the region. The random variable \(R\) is defined as the distance between \(C\) and \(O\). Find the probability density function of \(R\). Obtain a formula in terms of \(a\), \(R\) and \(t\) for the length of a chord through \(C\) that makes an acute angle of \(t\) with \(OC\). Show that as \(C\) varies (with \(t\) fixed), the expected length \(\mathrm{L}(t)\) of such chords is given by \[ \mathrm{L}(t) = \frac{4a(1-\cos^3 t)}{3\sin^2 t}\,. \] Show further that \[ \mathrm{L}(t) = \frac{4a}{3}\left(\cos t + \tfrac{1}{2}\sec^2(\tfrac{1}{2}t)\right). \]
3. The random variable \(T\) is uniformly distributed on \([0, \frac{1}{2}\pi]\). Find the expected value of \(\mathrm{L}(T)\).

The famous film star Birkhoff Maclane is sunning herself by the side of her enormous circular swimming pool (with centre \(O\)) at a point \(A\) on its circumference. She wants a drink from a small jug of iced tea placed at the diametrically opposite point \(B\). She has three choices:

1. to swim directly to \(B\).
2. to choose \(\theta\) with \(0<\theta<\pi,\) to run round the pool to a point \(X\) with \(\angle AOX=\theta\) and then to swim directly from \(X\) to \(B\).
3. to run round the pool from \(A\) to \(B\).

She can run \(k\) times as fast as she can swim and she wishes to reach her tea as fast as possible. Explain, with reasons, which of (i), (ii) and (iii) she should choose for each value of \(k\). Is there one choice from (i), (ii) and (iii) she will never take whatever the value of \(k\)?