Problems

A men's endurance competition has an unlimited number of rounds. In each round, a competitor has, independently, a probability \(p\) of making it through the round; otherwise, he fails the round. Once a competitor fails a round, he drops out of the competition; before he drops out, he takes part in every round. The grand prize is awarded to any competitor who makes it through a round which all the other remaining competitors fail; if all the remaining competitors fail at the same round the grand prize is not awarded. If the competition begins with three competitors, find the probability that:

1. all three drop out in the same round;
2. two of them drop out in round \(r\) (with \(r \ge 2\)) and the third in an earlier round;
3. the grand prize is awarded.

Show that, if \(n>0\,\), then $$ \int_{e^{1/n}}^\infty\,{{\ln x} \over {x^{n+1}}}\,\d x = {2 \over {n^2\e}}\;. $$ You may assume that \(\ds \frac{\ln x} x \to 0\;\) as \(x\to\infty\,\). Explain why, if \(1 < a < b\,\), then $$ \int_b^\infty\,{{\ln x} \over {x^{n+1}}}\,\d x < \int_a^\infty\,{{\ln x} \over {x^{n+1}}}\,\d x\;. $$ Deduce that $$ \sum_{n=1}^{N}{1 \over n^2} < {\e \over 2}\int_{\e^{1/N}}^{\infty} \left({1-x^{-N}} \over {x^2-x}\right) \ln x\,\d x\;, $$ where \(N\,\) is any integer greater than \(1\).

The life of a certain species of elementary particles can be described as follows. Each particle has a life time of \(T\) seconds, after which it disintegrates into \(X\) particles of the same species, where \(X\) is a random variable with binomial distribution \(\mathrm{B}(2,p)\,\). A population of these particles starts with the creation of a single such particle at \(t=0\,\). Let \(X_n\) be the number of particles in existence in the time interval \(nT < t < (n+1)T\,\), where \(n=1\,\), \(2\,\), \(\ldots\). Show that \(\P(X_1=2 \mbox { and } X_2=2) = 6p^4q^2\;\), where \(q=1-p\,\). Find the possible values of \(p\) if it is known that \(\P(X_1=2 \vert X_2=2) =9/25\,\). Explain briefly why \(\E(X_n) =2p\E(X_{n-1})\) and hence determine \(\E(X_n)\) in terms of \(p\). Show that for one of the values of \(p\) found above \(\lim_{n \to \infty}\E(X_n) = 0\) and that for the other \(\lim_{n \to \infty}\E(X_n) = + \infty\,\).

I borrow \(C\) pounds at interest rate \(100\alpha \,\%\) per year. The interest is added at the end of each year. Immediately after the interest is added, I make a repayment. The amount I repay at the end of the \(k\)th year is \(R_k\) pounds and the amount I owe at the beginning of \(k\)th year is \(C_k\) pounds (with \(C_1=C\)). Express \(C_{n+1}\) in terms of \(R_k\) (\(k= 1\), \(2\), \(\ldots\), \(n\)), \(\alpha\) and \(C\) and show that, if I pay off the loan in \(N\) years with repayments given by \(R_k= (1+\alpha)^kr\,\), where \(r\) is constant, then \(r=C/N\,\). If instead I pay off the loan in \(N\) years with \(N\) equal repayments of \(R\) pounds, show that \[ \frac R C = \frac{\alpha (1+\alpha)^{N} }{(1+\alpha)^N-1} \;, \] and that \(R/C\approx 27/103\) in the case \(\alpha =1/50\), \(N=4\,\).

In order to get money from a cash dispenser I have to punch in an identification number. I have forgotten my identification number, but I do know that it is equally likely to be any one of the integers \(1\), \(2\), \ldots , \(n\). I plan to punch in integers in order until I get the right one. I can do this at the rate of \(r\) integers per minute. As soon as I punch in the first wrong number, the police will be alerted. The probability that they will arrive within a time \(t\) minutes is \(1-\e^{-\lambda t}\), where \(\lambda\) is a positive constant. If I follow my plan, show that the probability of the police arriving before I get my money is \[ \sum_{k=1}^n \frac{1-\e^{-\lambda(k-1)/r}}n\;. \] Simplify the sum. On past experience, I know that I will be so flustered that I will just punch in possible integers at random, without noticing which I have already tried. Show that the probability of the police arriving before I get my money is \[ 1-\frac1{n-(n-1)\e^{-\lambda/r}} \;. \]

A particle moves on a smooth triangular horizontal surface \(AOB\) with angle \(AOB = 30^\circ\). The surface is bounded by two vertical walls \(OA\) and \(OB\) and the coefficient of restitution between the particle and the walls is \(e\), where \(e < 1\). The particle, which is initially at point \(P\) on the surface and moving with velocity \(u_1\), strikes the wall \(OA\) at \(M_1\), with angle \(PM_1A = \theta\), and rebounds, with velocity \(v_1\), to strike the wall \(OB\) at \(N_1\), with angle \(M_1N_1B = \theta\). Find \(e\) and \(\displaystyle {v_1 \over u_1}\) in terms of \(\theta\). The motion continues, with the particle striking side \(OA\) at \(M_2\), \(M_3\), \( \ldots \) and striking side \(OB\) at \(N_2\), \(N_3\), \(\ldots \). Show that, if \(\theta < 60^\circ\,\), the particle reaches \(O\) in a finite time.

In a game, a player tosses a biased coin repeatedly until two successive tails occur, when the game terminates. For each head which occurs the player wins \(\pounds 1\). If \(E\) is the expected number of tosses of the coin in the course of a game, and \(p\) is the probability of a head, explain why \[ E = p \l 1 + E \r + \l 1 - p \r p \l 2 + E \r + 2 \l 1 - p \r ^2\,, \] and hence determine \(E\) in terms of \(p\). Find also, in terms of \(p\), the expected winnings in the course of a game. A second game is played, with the same rules, except that the player continues to toss the coin until \(r\) successive tails occur. Show that the expected number of tosses in the course of a game is given by the expression \(\displaystyle {1 - q^r \over p q^r}\,\), where \(q = 1 - p\).

Sketch the graph of the function \([x/N]\), for \(0 < x < 2N\), where the notation \([y]\) means the integer part of \(y\). (Thus \([2.9] = 2\), \ \([4]=4\).)

1. Prove that \[ \sum_{k=1}^{2N} (-1)^{[k/N]} k = 2N-N^2. \]
2. Let \[ S_N = \sum_{k=1}^{2N} (-1)^{[k/N]} 2^{-k}. \] Find \(S_N\) in terms of \(N\) and determine the limit of \(S_N\) as \(N\to\infty\).

\(N\) particles \(P_1\), \(P_2\), \(P_3\), \(\ldots\), \(P_N\) with masses \(m\), \(qm\), \(q^2m\), \(\ldots\) , \({q^{N-1}}m\), respectively, are at rest at distinct points along a straight line in gravity-free space. The particle \(P_1\) is set in motion towards \(P_2\) with velocity \(V\) and in every subsequent impact the coefficient of restitution is \(e\), where \(0 < e < 1\). Show that after the first impact the velocities of \(P_1\) and \(P_2\) are $$ {\left({{1-eq}\over{1+q}}\right)}V \mbox{ \ \ \ and \ \ \ } {\left({{1+e}\over{1+q}}\right)}V, $$ respectively. Show that if \(q \le e\), then there are exactly \(N-1\) impacts and that if \(q=e\), then the total loss of kinetic energy after all impacts have occurred is equal to $$ {1\over 2}{me}{\left(1-e^{N-1}\right)}{V^2}. $$

My bank pays \(\rho\%\) interest at the end of each year. I start with nothing in my account. Then for \(m\) years I deposit \(\pounds a\) in my account at the beginning of each year. After the end of the \(m\)th year, I neither deposit nor withdraw for \(l\) years. Show that the total amount in my account at the end of this period is \[\pounds a\frac{r^{l+1}(r^{m}-1)}{r-1}\] where \(r=1+{\displaystyle \frac{\rho}{100}}\). At the beginning of each of the \(n\) years following this period I withdraw \(\pounds b\) and this leaves my account empty after the \(n\)th withdrawal. Find an expression for \(a/b\) in terms of \(r\), \(l\), \(m\) and \(n\).

I have a bag initially containing \(r\) red fruit pastilles (my favourites) and \(b\) fruit pastilles of other colours. From time to time I shake the bag thoroughly and remove a pastille at random. (It may be assumed that all pastilles have an equal chance of being selected.) If the pastille is red I eat it but otherwise I replace it in the bag. After \(n\) such drawings, I find that I have only eaten one pastille. Show that the probability that I ate it on my last drawing is \[\frac{(r+b-1)^{n-1}}{(r+b)^{n}-(r+b-1)^{n}}.\]

The value \(V_N\) of a bond after \(N\) days is determined by the equation $$ V_{N+1} = (1+c) V_{N} -d \qquad (c>0, \ d>0), $$ where \(c\) and \(d\) are given constants. By looking for solutions of the form \(V_T= A k^T + B\) for some constants \(A,B\) and \(k\), or otherwise, find \(V_N\) in terms of \(V_0\). What is the solution for \(c=0\)? Show that this is the limit (for fixed \(N\)) as \(c\rightarrow 0\) of your solution for \(c>0\).

For each positive integer \(n\), let \begin{align*} a_n&=\frac1{n+1}+\frac1{(n+1)(n+2)}+\frac1{(n+1)(n+2)(n+3)}+\cdots;\\ b_n&=\frac1{n+1}+\frac1{(n+1)^2}+\frac1{(n+1)^3}+\cdots. \end{align*}

1. Evaluate \(b_n\).
2. Show that \(0
3. Deduce that \(a_n=n!{\rm e}-[n!{\rm e}]\) (where \([x]\) is the integer part of \(x\)).
4. Hence show that \(\mathrm{e}\) is irrational.

1. By using the formula for the sum of a geometric series, or otherwise, express the number \(0.38383838\ldots\) as a fraction in its lowest terms.
2. Let \(x\) be a real number which has a recurring decimal expansion \[ x=0\cdot a_{1}a_{2}a_{2}\cdots, \] so that there exists positive integers \(N\) and \(k\) such that \(a_{n+k}=a_{n}\) for all \(n>N.\) Show that \[ x=\frac{b}{10^{N}}+\frac{c}{10^{N}(10^{k}-1)}\,, \] where \(b\) and \(c\) are integers to be found. Deduce that \(x\) is rational.

A biased coin, with a probability \(p\) of coming up heads and a probability \(q=1-p\) of coming up tails, is tossed repeatedly. Let \(A\) be the event that the first run of \(r\) successive heads occurs before the first run of \(s\) successive tails. If \(H\) is the even that on the first toss the coin comes up heads and \(T\) is the event that it comes up tails, show that \begin{alignat*}{1} \mathrm{P}(A|H) & =p^{\alpha}+(1-p^{\alpha})\mathrm{P}(A|T),\\ \mathrm{P}(A|T) & =(1-q^{\beta})\mathrm{P}(A|H), \end{alignat*} where \(\alpha\) and \(\beta\) are to be determined. Use these two equations to find \(\mathrm{P}(A|H),\) \(\mathrm{P}(A|T),\) and hence \(\mathrm{P}(A).\)