A Bernoulli trial results in success with probability \(p\) or failure with probability \(1-p\). If \(X\) is the number of successes in \(n\) independent Bernoulli trials show that \(E(X) = np\) and \(\text{var}(X) = npq\). We wish to estimate by means of a random sample the proportion \(p\) of the population of a certain large city who are cat lovers. How large should the sample be if we wish to be 95\% certain that the error in estimating \(p\) will be less than 0.01?
In tennis, players serve in alternate games and a set is won when one player has won six games, except that whenever a score of five games all is reached play continues until one player has a lead of two games. A player is leading by four games to two. His chance of winning a game when he serves is \(\frac{3}{4}\) and his chance of winning when his opponent serves is \(\frac{1}{4}\). What is the probability that he will win the set?
Initially a machine is in good running order but is subsequently liable to break down. As soon as a breakdown occurs repairs begin. If the machine is in good order at time \(t\) then the probability that a breakdown occurs in a small interval \((t, t + dt)\) is \(\alpha dt\), and if it is under repair at time \(t\) the probability that the repair is completed in time \((t, t + dt)\) is \(\beta dt\). Let \(p(t)\) be the probability that the machine is under repair at time \(t\). Write down an equation relating \(p(t + dt)\) to \(p(t)\) and hence show that \(p(t)\) is $$\frac{\alpha}{\alpha + \beta}\{1 - \exp[- (\alpha + \beta)t]\}.$$
Solution: Here we are assuming that the interval is small enough that there aren't multiple events happening. \begin{align*} && \underbrace{p(t+dt)}_{\text{P under repair at time \(t+dt\)}} &= \underbrace{p(t)}_{\text{P under repair at time \(t\)}}\underbrace{(1-\beta)dt}_{\text{P not fixed in time}} + \underbrace{(1-p(t))}_{\text{P not under repair}} \underbrace{\alpha dt}_{\text{P breaks}} \\ &&&= p(t)(1-\beta-\alpha)dt + \alpha dt \\ \Rightarrow && \frac{p(t+dt)-p(t)}{dt} &= -(\alpha+\beta)p(t) + \alpha \\ \Rightarrow && \frac{\d p}{\d t} &= -(\alpha+\beta)p(t) + \alpha \\ \Rightarrow && p(t) &= A\exp(-(\alpha+\beta)t) + B \\ \Rightarrow && 0 &= -(\alpha+\beta)B + \alpha \\ \Rightarrow && B &= \frac{\alpha}{\alpha+\beta} \\ \Rightarrow && 0 &= A + \frac{\alpha}{\alpha+\beta} \\ \Rightarrow && A &= - \frac{\alpha}{\alpha+\beta} \\ \Rightarrow && p(t) &= \frac{\alpha}{\alpha+\beta} \left ( 1 - \exp(-(\alpha+\beta)t)\right) \end{align*}
You are given a coin and told that it is equally likely to be one which has probability 0.8 of coming down heads if tossed, or one which has probability 0.2 of coming down heads if tossed. You must decide which coin it is. Choosing wrongly will cost you nothing, but choosing correctly will gain you 2 units if it is really the former coin and 1 unit if it is really the latter.
Solution:
Six equal light rods are jointed together to form a regular tetrahedron \(ABCD\). Equal and opposite forces \(F\) are applied at the midpoints of \(AB\) and \(CD\) directed towards the centre of the tetrahedron. Calculate the tensions or thrusts in the rods.
Establish the equation of motion of a simple pendulum of length \(l\) in terms of the angle \(\theta\) that the pendulum makes with the upward vertical. Deduce the equation expressing the conservation of energy. Find \(\theta\) as a function of \(t\) given that, at time \(t = 0\), \(\theta = \pi\) and the kinetic energy is \(2mgl\); and show that the time taken for the pendulum to reach a position within a small angle \(\alpha\) of the upward vertical is approximately \(\sqrt{(l/g)} \ln (4/\alpha)\). $$\left[\int \text{cosec } x \, dx = \ln \tan \frac{1}{2}x.\right]$$
One edge of a uniform cube lies against a smooth vertical wall and another edge rests on a horizontal surface with coefficient of friction \(\mu\). The face between these two edges is inclined at an angle \(\phi\) to the wall. Find the range of values of \(\phi\) for which equilibrium is possible. What happens if the wall is rough but the horizontal surface is smooth?
Two rocket bases \(A\), equipped with rockets that travel at a fixed speed \(M/\tau\) (\(M > 1\)), lie due west of a similar base \(B\). They are separated by a distance \((1 + \sqrt{3})d\). An aircraft flies due west with uniform velocity at height \(d\) at time \(t = -\tau\), in directions 45° and 300° from North (measured clockwise) as seen from \(A\) and \(B\) respectively. At what instant \(t\) crosses \(AB\) at a point distant \(d\) east from \(A\). How soon can the aircraft be intercepted by a rocket fired at \(t = 0\)?
A mountaineer falls over a cliff. He is attached to a rope which, providentially, catches so that he just touches the ground at the foot of the cliff. Find the height of the cliff and the time taken for the mountaineer to reach the ground (in terms of his mass, the length of the unstretched rope and the elastic modulus of the rope).
A sphere moving with velocity \(\mathbf{u}_1 = a_1\mathbf{u}\) collides with a similar sphere moving with velocity \(\mathbf{u}_2 = a_2\mathbf{u}\). Momentum and energy are conserved in the collision, after which the spheres have velocities \(\mathbf{v}_1\) and \(\mathbf{v}_2\) respectively. Show that if \(\mathbf{v}_1\) and \(\mathbf{v}_2\) are mutually perpendicular then one of the spheres must initially have been stationary. Is the converse true? If both spheres have the same speed \(c|\mathbf{u}|\) after the collision, show that \(c^2 \cos \theta = a_1 a_2\), where \(\theta\) is the angle between \(\mathbf{v}_1\) and \(\mathbf{v}_2\).