The mountain villages \(A\), \(B\), \(C\), \(D\) lie at the vertices of a tetrahedron, and each pair of villages is joined by a road. After a snowfall the probability that any road is blocked is \(p\), and is independent of the conditions on any other road. Find the probability that it is possible to travel from any village to any other village by some route after snowfall. In the case \(p = \frac{1}{2}\) show that this probability is \(\frac{19}{32}\).
\(A\) and \(B\) play the following game. \(A\) throws two unbiased four-sided dice (each has the numbers 1 to 4 on its sides), and notes the total \(Y\). \(B\) tries to guess this number, and guesses \(X\). If \(B\) guesses correctly he wins \(X^2\) pounds, and if he is wrong he loses \(\frac{1}{2}X\) pounds. (a) Show that \(B\)s average gain if he always guesses 8 is \(\frac{1}{4}\). (b) He decides that he will always guess the same value of \(X\). Which value of \(X\) would you advise him to choose, and what is his average gain in this case?
The lifetime in days, \(X\), of a safety component in a chemical plant is given by the negative exponential distribution \begin{align} P(X \leq t) = 1 - e^{-\lambda t} \text{ for } t \geq 0 \end{align} Find the mean lifetime of the component. The component is checked at 8 o'clock every morning, and if faulty is replaced immediately. Let \(Y\) be the length of time, in days, between the component failing and being replaced. Show that the probability that the component fails on the \(n\)th day and is replaced within \(24y\) hours, where \(0 \leq y \leq 1\), is \((e^{\lambda y} - 1)e^{-\lambda n}\) for \(n = 1, 2, ...\). Hence prove that \begin{align} P(Y \leq y) = \frac{e^{\lambda y} - 1}{e^{\lambda} - 1} \end{align} and calculate the mean of \(Y\).
Let \(X_1, ..., X_m\) be independent normally distributed random variables, with mean \(\mu\) and variance \(\sigma^2\). Let \(X > 0\), and let \(Y\) be the number of observations falling in the range \((a-X, \mu+X)\). Give an expression for \(P(Y = r)\) for \(r = 0, 1, ..., m\). If \(\alpha = \frac{1}{2}\) and \(m = 10\), what is \(P(Y \leq 2)\)? (You may leave your answer in a form suitable for calculation.)
A uniform rod \(AB\) of length \(l\) lies on a rough horizontal table. A string is attached to the rod at \(B\) and is pulled in a horizontal direction perpendicular to the rod. Show that, as the tension in the string is gradually increased, the rod begins to turn about a point whose distance from \(A\) is \(l(1 - 1/\sqrt{3})\), and find the value of the tension when that occurs, in terms of the weight of the rod and the coefficient of friction between the rod and the table.
An inclined plane makes an angle \(\alpha\) with the horizontal. A small, perfectly elastic sphere is projected up the plane at an angle of elevation \(\beta\) relative to the plane. Its second bounce occurs at the point of projection. Show that \(2 \tan \alpha \tan \beta = 1\).
A particle moves in the \(x\)-\(y\) plane with the following equations of motion: \begin{align} \ddot{x} = y\dot{d}, \quad \ddot{y} = c - x\dot{d} \end{align} where \(c\) and \(d\) are constant. Show that the quantity \(\frac{1}{2}(\dot{x}^2 + \dot{y}^2) - cy\) is constant. At \(t = 0\) the initial conditions are \(x = 0\), \(y = 0\), \(\dot{x} = Q\) and \(\dot{y} = 0\). Show that the motion is along the \(x\)-axis if and only if \(Q\) has a certain value, which is to be determined.
A small body of mass \(M\) is moving with velocity \(v\) along the axis of a long, smooth, fixed, circular cylinder of radius \(L\). An internal explosion splits the body into two spherical fragments, with masses \(qM\) and \((1-q)M\), where \(q \leq \frac{1}{2}\). After bouncing elastically off the cylinder (one bounce each) the fragments collide and coalesce. The collision occurs a time \(5L/v\) after the explosion and at a point \(\frac{3}{4}L\) from the axis. Show that \(q = \frac{3}{8}\). Find the energy imparted to the fragments by the explosion, and find the velocity after coalescence. The effect of gravity may be neglected.
The banks of a straight river are given by \(x = 0\) and \(x = a\) in a horizontal rectangular coordinate system \((x, y)\). The water flows in the positive \(y\)-direction with a speed \(3ux(a-x)/a^2\) which depends on the distance \(x\) from the bank. An otter which swims at a steady speed \(u\) starts from the coordinate origin and swims at a constant angle \(\theta\) to the current. Evaluate \(dy/dx\) for its motion and hence find \(y\) as a function of \(x\). If it arrives at the far bank at the point \((a, 0)\) directly opposite its starting point, show that \(\theta = \frac{2\pi}{3}\). For this case find also the values of \(x\) for which \(|y|\) is maximum.
A particle of mass \(m\) is projected vertically upwards in a medium which resists the motion with a force \(mk v^2\), where \(v\) is the speed of the particle. Show that it reaches its greatest height in a time less than \(\pi/2\sqrt{(kg)}\), where \(g\) is the acceleration of gravity. If its speed of projection is \(u\), find its speed when it returns to ground level.