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2001 Paper 1 Q11
A smooth cylinder with circular cross-section of radius \(a\) is held with its axis horizontal. A~light elastic band of unstretched length \(2\pi a\) and modulus of elasticity \(\lambda\) is wrapped round the circumference of the cylinder, so that it forms a circle in a plane perpendicular to the axis of the cylinder. A particle of mass \(m\) is then attached to the rubber band at its lowest point and released from rest.
- Given that the particle falls to a distance \(2a\) below the below the axis of the cylinder, but no further, show that \[ \lambda = \frac{9\pi m g}{(3\sqrt3-\pi)^2} \;. \]
- Given instead that the particle reaches its maximum speed at a distance \(2a\) below the axis of the cylinder, find a similar expression for \(\lambda\)\,.
2001 Paper 1 Q12
Four students, Arthur, Bertha, Chandra and Delilah, exchange gossip. When Arthur hears a rumour, he tells it to one of the other three without saying who told it to him. He decides whom to tell by choosing at random amongst the other three, omitting the ones that he knows have already heard the rumour. When Bertha, Chandra or Delilah hear a rumour, they behave in exactly the same way (even if they have already heard it themselves). The rumour stops being passed round when it is heard by a student who knows that the other three have already heard it. Arthur starts a rumour and tells it to Chandra. By means of a tree diagram, or otherwise, show that the probability that Arthur rehears it is \(3/4\). Find also the probability that Bertha hears it twice and the probability that Chandra hears it twice.
Solution: Without loss of generality, \(C\) will tell \(B\) about the rumour. If \(B\) tells \(D\) then \(D\) can either tell \(A\) or \(C\) at which point either \(A\) is told or the rumour stops spreading.
2001 Paper 1 Q13
Four students, one of whom is a mathematician, take turns at washing up over a long period of time. The number of plates broken by any student in this time obeys a Poisson distribution, the probability of any given student breaking \(n\) plates being \(\e^{-\lambda} \lambda^n/n!\) for some fixed constant \(\lambda\), independent of the number of breakages by other students. Given that five plates are broken, find the probability that three or more were broken by the mathematician.
Solution: Let \(X\) be the number of plates broken by the mathematician and \(Y\) by the other student. Then \(X \sim Po(\lambda), Y \sim Po(3\lambda)\) and \(X+Y \sim Po(4\lambda)\) \begin{align*} && \mathbb{P}(X = k | X+Y = n) &= \frac{\mathbb{P}(X = k, Y = n-k)}{\mathbb{P}(X+Y=n)} \\ &&&= \frac{e^{-\lambda} \lambda^k/k! \cdot e^{-3\lambda} (4\lambda)^{n-k}/(n-k)!}{e^{-4\lambda}(4\lambda)^n/n!} \\ &&&= \binom{n}{k} \left ( \frac{1}{4} \right)^k \left ( \frac{3}{5} \right)^{n-k} \end{align*} Therefore \(X | X+Y = n \sim Binomial(n, \tfrac14)\) \begin{align*} \mathbb{P}(X \geq 3 | X + Y = n) &= \binom{5}{3} \frac{3^2}{4^5} + \binom{5}{4} \frac{3}{4^5} + \binom{5}{5} \frac{1}{4^5} \\ &= \frac{1}{4^5} \left ( 90+ 15 + 1 \right) \\ &= \frac{106}{4^5} = \frac{53}{512} \approx \frac1{10} \end{align*}
2001 Paper 1 Q14
On the basis of an interview, the \(N\) candidates for admission to a college are ranked in order according to their mathematical potential. The candidates are interviewed in random order (that is, each possible order is equally likely).
- Find the probability that the best amongst the first \(n\) candidates interviewed is the best overall.
- Find the probability that the best amongst the first \(n\) candidates interviewed is the best or second best overall.
Solution:
- The probability the best person falls in the first \(n\) is \(\frac{n}{N}\)
- The probability the best two people do not fall in the first \(n\) candidates is \begin{align*} && 1-P &= \frac{\binom{N-2}{n}}{\binom{N}{n}} \\ &&&= \frac{(N-2)(N-3)\cdots(N-2-n+1)}{n!} \frac{n!}{N(N-1)(N-2) \cdots (N-n+1)} \\ &&&= \frac{(N-n)(N-n-1)}{N(N-1)} \\ \Rightarrow && P &= 1- \frac{(N-n)(N-n-1)}{N(N-1)} \\ &&&= \frac{N(N-1) - N(N-1)+n(N-n-1)+Nn}{N(N-1)} \\ &&&= \frac{n(2N-n-1)}{N(N-1)} \end{align*}