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1995 Paper 1 Q11
D: 1500.0 B: 1500.0

Two identical uniform cylinders, each of mass \(m,\) lie in contact with one another on a horizontal plane and a third identical cylinder rests symmetrically on them in such a way that the axes of the three cylinders are parallel. Assuming that all the surfaces in contact are equally rough, show that the minimum possible coefficient of friction is \(2-\sqrt{3}.\)

Solution:

TikZ diagram
First observe that many forces are equal by symmetry. Also notice that \(A\) and \(B\) are trying to roll in opposite directions, therefore there is no friction between \(A\) and \(B\). Considering the system as a whole \(R_1 = \frac32 mg\). \begin{align*} \text{N2}(\uparrow,C): && 0 &= -mg +2R_3\cos 30^{\circ} + 2F_{CA} \cos 60^{\circ} \\ \Rightarrow && mg &= \sqrt{3}R_3 + F_{CA} \\ \\ \text{N2}(\uparrow, A): && 0 &= -mg + \frac32mg-R_3\cos 30^{\circ} -F_{AC} \cos 60^\circ \\ \Rightarrow && mg &= \sqrt{3}R_3+F_{AC} \\ \text{N2}(\rightarrow, A): && 0 &= F_{AC}\cos 30^{\circ}+F_1-R_3 \cos 60^\circ -R_2 \\ \Rightarrow && 0 &=F_{AC}\sqrt{3}+2F_1-R_3-2R_2 \\ \overset{\curvearrowleft}{A}: && 0 &= F_1 - F_{AC} \end{align*} Since \(F_1 = F_{AC} = F_{CA}\) we can rewrite everything in terms of \(F= F_1\) so. \begin{align*} && mg &= \sqrt{3}R_3 + F \\ && 0 &= (2+\sqrt{3})F -R_3-2R_2 \\ \Rightarrow && (2+\sqrt3)F &\geq R_3 \\ \Rightarrow && F &\geq (2-\sqrt{3})R_3 \\ \Rightarrow && \mu & \geq 2 - \sqrt{3} \end{align*}

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1995 Paper 1 Q12
D: 1500.0 B: 1501.9

A school has \(n\) pupils, of whom \(r\) play hocket, where \(n\geqslant r\geqslant2.\) All \(n\) pupils are arranged in a row at random.

  1. What is the probability that there is a hockey player at each end of the row?
  2. What is the probability that all the hockey players are standing together?
  3. By considering the gaps between the non-hockey-players, find the probability that no two hockey players are standing together, distinguishing between cases when the probability is zero and when it is non-zero.

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1995 Paper 1 Q13
D: 1500.0 B: 1484.0

A scientist is checking a sequence of microscope slides for cancerous cells, marking each cancerous cell that she detects with a red dye. The number of cancerous cells on a slide is random and has a Poisson distribution with mean \(\mu.\) The probability that the scientist spots any one cancerous cell is \(p\), and is independent of the probability that she spots any other one.

  1. Show that the number of cancerous cells which she marks on a single slide has a Poisson distribution of mean \(p\mu.\)
  2. Show that the probability \(Q\) that the second cancerous cell which she marks is on the \(k\)th slide is given by \[ Q=\mathrm{e}^{-\mu p(k-1)}\left\{ (1+k\mu p)(1-\mathrm{e}^{-\mu p})-\mu p\right\} . \]

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1995 Paper 1 Q14
D: 1516.0 B: 1531.3

  1. Find the maximum value of \(\sqrt{p(1-p)}\) as \(p\) varies between \(0\) and \(1\).
  2. Suppose that a proportion \(p\) of the population is female. In order to estimate \(p\) we pick a sample of \(n\) people at random and find the proportion of them who are female. Find the value of \(n\) which ensures that the chance of our estimate of \(p\) being more than \(0.01\) in error is less than 1\%.
  3. Discuss how the required value of \(n\) would be affected if (a) \(p\) were the proportion of people in the population who are left-handed; (b) \(p\) were the proportion of people in the population who are millionaires.

Solution:

  1. \(\,\) \begin{align*} && \sqrt{p(1-p)} &= \sqrt{p-p^2} \\ &&&= \sqrt{\tfrac14-(\tfrac12-p)^2} \\ &&&\leq \sqrt{\tfrac14} = \tfrac12 \end{align*} Therefore the maximum is \(\tfrac12\) when \(p=\frac12\)
  2. Notice that our estimate \(\hat{p}\) will (for large \(n\)) be follow a normal distribution \(N(p, pq/n)\) by either the normal approximation to the binomial or central limit theorem. We would like \(0.01 > \mathbb{P}\left ( |\hat{p}-p| < 0.01 \right)\) or in other words \begin{align*} && 0.01 &> \mathbb{P}\left ( |\hat{p}-p| > 0.01 \right) \\ &&&=\mathbb{P}\left ( |\sqrt{\frac{pq}{n}}Z+p-p| > 0.01 \right) \\ &&&= \mathbb{P} \left (|Z|>\frac{0.01\sqrt{n}}{\sqrt{pq}}\right) \end{align*} therefore we need \(\frac{0.01\sqrt{n}}{\sqrt{pq}}> 2.58 \Rightarrow n > 258^2 pq \approx 2^{14} \approx 16\,000\), where we are using \(pq = \frac14\) as the worst case possibility and \(258 \approx 256 = 2^8\)
  3. If we were looking at when we are looking at left handed people (maybe ~\(10\%\), we would be looking at \(pq = \frac{9}{100}\) so we need a smaller sample). If we are looking at millionaires (an even smaller again percentage), we would need an even smaller sample. This is surprising since you would expect you would need a larger sample to accurately gauge smaller proportions. However, this surprise can be resolved by considering that this is an absolute error. For smaller values the relative error is larger, but the absolute error is smaller.

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