Seven sunbathers are positioned at equal intervals along a straight shoreline. Each stares fixedly at a nearest neighbour, choosing a neighbour at random if a choice is available. Show that the expected number of unobserved sunbathers is \(\frac{3}{4}\).
A jar contains \(r\) red, \(b\) blue and \(w\) white sweets. A greedy child picks out sweets one by one at random and eats them, until only sweets of a single colour remain. Show, by induction or otherwise, that the probability that only red sweets remain is \(\frac{r}{r + b + w}\).
Solution: Imagine we continue pulling out sweets. Then the last sweet to be removed will be the same colour as all the other sweets which are the same colour. Therefore the question is equivalent to "what is the probability the last sweet is red", but since all possible orders of sweets are equally likely, this is just \(\frac{r}{r+b+w}\)
A chocolate orange consists of a sphere of smooth uniform chocolate of mass \(M\) and radius \(a\), sliced into segments by planes through its axis. It stands on a horizontal table with its axis vertical, and it is held together only by a narrow ribbon around its equator. Show that the tension in the ribbon is at least \(\frac{3}{8\pi}Mg\). [You may assume that the centre of mass of a segment of angle \(\theta\) is at a distance \((3a/2\pi)\sin(\theta/2)\) from the axis.]
A light spring has natural length \(a\) and is such that when compressed a distance \(x\) it produces a force of magnitude \(kx\). It joins two particles of masses \(m_1\) and \(m_2\). The spring is compressed a distance \(b\) and the system released from rest on a smooth table, so that the particles move in a straight line. Find the positions of the particles at time \(t\) later.
Concorde flies the distance \(d\) from London to New York in an average time \(t_1\) and makes the return journey in an average time \(t_2\) where \(t_2 < t_1\). Assuming that the earth is flat and that Concorde flies at a uniform speed \(V\) in still air, find the speed of the prevailing wind and the angle it makes with the straight line joining New York to London.
Particles of mud are thrown off the tyres of the wheels of a cart travelling at constant speed \(V\). Neglecting air resistance, show that a particle which leaves the ascending part of a tyre at a point above the hub will be thrown clear of the wheel provided its height above the hub at the instant when it leaves the tyre is greater than \(a^2/V^2\), where \(a\) is the radius of the tyre.