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2014 Paper 2 Q13
D: 1600.0 B: 1469.5
probability uniform distribution discrete random variable expectation birthday problem summation identity proof

A random number generator prints out a sequence of integers \(I_1, I_2, I_3, \dots\). Each integer is independently equally likely to be any one of \(1, 2, \dots, n\), where \(n\) is fixed. The random variable \(X\) takes the value \(r\), where \(I_r\) is the first integer which is a repeat of some earlier integer. Write down an expression for \(\mathbb{P}(X=4)\).

  1. Find an expression for \(\mathbb{P}(X=r)\), where \(2\le r\le n+1\). Hence show that, for any positive integer \(n\), \[ \frac 1n + \left(1-\frac1n\right) \frac 2 n + \left(1-\frac1n\right)\left(1-\frac2n\right) \frac3 n + \cdots \ = \ 1 \,. \]
  2. Write down an expression for \(\mathbb{E}(X)\). (You do not need to simplify it.)
  3. Write down an expression for \(\mathbb{P}(X\ge k)\).
  4. Show that, for any discrete random variable \(Y\) taking the values \(1, 2, \dots, N\), \[ \mathbb{E}(Y) = \sum_{k=1}^N \mathbb{P}(Y\ge k)\,. \] Hence show that, for any positive integer \(n\), \[ \left(1-\frac{1^2}n\right) + \left(1-\frac1n\right)\left(1-\frac{2^2}n\right) + \left(1-\frac1n\right)\left(1-\frac{2}n\right)\left(1-\frac{3^2}n\right) + \cdots \ = \ 0. \]

Solution: \begin{align*} && \mathbb{P}(X > 4) &= 1 \cdot \frac{n-1}{n} \cdot \frac{n-2}{n} \cdot \frac{n-3}{n} \\ && \mathbb{P}(X > 3) &= 1 \cdot \frac{n-1}{n} \cdot \frac{n-2}{n} \\ \Rightarrow && \mathbb{P}(X =4) &= \mathbb{P}(X > 3) - \mathbb{P}(X > 4) \\ &&&= \frac{(n-1)(n-2)}{n^2} \left (1 - \frac{n-3}{n} \right) \\ &&&= \frac{3(n-1)(n-2)}{n^3} \end{align*}

  1. Notice that \begin{align*} && \mathbb{P}(X > r) &= \frac{n-1}{n} \cdots \frac{n-r+1}{n} \\ \Rightarrow && \mathbb{P}(X = r) &= \frac{n-1}{n} \cdots \frac{n-r+2}{n} \left (1 - \frac{n-r+1}{n} \right) \\ &&&= \frac{(n-1)\cdots(n-r+2)(r-1)}{n^{r-1}} \\ &&&= \left (1 - \frac{1}n \right)\left (1 - \frac{2}{n} \right) \cdots \left (1 - \frac{r-2}{n} \right) \frac{r-1}{n} \\ \Rightarrow && 1 &= \sum \mathbb{P}(X = r) \\ &&&= \sum_{r=2}^{n+1} \mathbb{P}(X = r) \\ &&&= \frac 1n + \left(1-\frac1n\right) \frac 2 n + \left(1-\frac1n\right)\left(1-\frac2n\right) \frac3 n + \cdots \end{align*}
  2. \(\,\) \begin{align*} \mathbb{E}(X) &= \sum_{r=2}^{n+1} r\cdot\mathbb{P}(X = r) \\ &= \frac 2n + \left(1-\frac1n\right) \frac {2\cdot3} n + \left(1-\frac1n\right)\left(1-\frac2n\right) \frac{3\cdot4} n + \cdots \end{align*}
  3. \(\displaystyle \mathbb{P}(X \geq k) = \frac{n-1}{n} \cdots \frac{n-r+2}{n}\)
  4. \(\,\) \begin{align*} && \mathbb{E}(Y) &= \sum_{r=1}^N r \cdot \mathbb{P}(Y = r) \\ &&&= \sum_{r=1}^N \sum_{j=1}^r \mathbb{P}(Y = r) \\ &&&= \sum_{j=1}^N \sum_{r=j}^N \mathbb{P}(Y=r) \\ &&&= \sum_{j=1}^N \mathbb{P}(Y \geq j) \end{align*} Let \(P_k = \left(1-\frac1n\right)\left(1-\frac2n\right) \cdots \left(1-\frac1n\right)\left(1-\frac{k}n\right) \) \begin{align*} && \mathbb{E}(X) &= P_1 \frac{1 \cdot 2 }{n} + P_2 \cdot \frac{2 \cdot 3}{n} + \cdots + P_k \cdot \frac{k(k+1)}{n} + \cdots \\ && &= \sum_{k=1}^{n} \frac{k^2}{n}P_k + \sum_{k=1}^{n} \frac{k}{n}P_k \\ && \text{Using the identity } & \frac{k}{n}P_k = \frac{k}{n} \prod_{i=1}^{k-1} \left(1-\frac{i}{n}\right) = P_k - P_{k+1}: \\ && \sum_{k=1}^{n} \frac{k}{n}P_k &= (P_1 - P_2) + (P_2 - P_3) + \cdots + (P_n - P_{n+1}) \\ && &= P_1 - P_{n+1} = 1 - 0 = 1 \\ \\ \Rightarrow && \mathbb{E}(X) &= \sum_{k=1}^{n} \frac{k^2}{n}P_k + 1 \\ && &= \mathbb{P}(X \geq 1) + \mathbb{P}(X \geq 2) + \mathbb{P}(X \geq 3) + \cdots \\ && &= 1 + P_1 + P_2 + P_3 + \cdots \\ && &= 1 + \sum_{k=1}^{n} P_k \\ \\ \Rightarrow && 1 + \sum_{k=1}^{n} P_k &= \sum_{k=1}^{n} \frac{k^2}{n}P_k + 1 \\ \Rightarrow && \sum_{k=1}^{n} P_k &= \sum_{k=1}^{n} \frac{k^2}{n}P_k \\ \Rightarrow && 0 &= \sum_{k=1}^{n} P_k \left( 1 - \frac{k^2}{n} \right) \end{align*}

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2014 Paper 3 Q12
D: 1700.0 B: 1500.0
probability cumulative distribution function probability density function transformation of variables normal distribution mode median expectation inequality lognormal distribution completing the square

The random variable \(X\) has probability density function \(f(x)\) (which you may assume is differentiable) and cumulative distribution function \(F(x)\) where \(-\infty < x < \infty \). The random variable \(Y\) is defined by \(Y= \e^X\). You may assume throughout this question that \(X\) and \(Y\) have unique modes.

  1. Find the median value \(y_m\) of \(Y\) in terms of the median value \(x_m\) of \(X\).
  2. Show that the probability density function of \(Y\) is \(f(\ln y)/y\), and deduce that the mode \(\lambda\) of \(Y\) satisfies \(\f'(\ln \lambda) = \f(\ln \lambda)\).
  3. Suppose now that \(X \sim {\rm N} (\mu,\sigma^2)\), so that \[ f(x) = \frac{1}{\sigma \sqrt{2\pi}\,} \e^{-(x-\mu)^2/(2\sigma^2)} \,. \] Explain why \[\frac{1}{\sigma \sqrt{2\pi}\,} \int_{-\infty}^{\infty}\e^{-(x-\mu-\sigma^2)^2/(2\sigma^2)} \d x = 1 \] and hence show that \( \E(Y) = \e ^{\mu+\frac12\sigma^2}\).
  4. Show that, when \(X \sim {\rm N} (\mu,\sigma^2)\), \[ \lambda < y_m < \E(Y)\,. \]

Solution:

  1. \begin{align*} && \frac12 &= \mathbb{P}(X \leq x_m) \\ \Leftrightarrow && \frac12 &= \mathbb{P}(e^X \leq e^{x_m} = y_m) \end{align*} Therefore the median is \(y_m = e^{x_m}\)
  2. \begin{align*} && \mathbb{P}(Y \leq y) &= \mathbb{P}(e^X \leq y) \\ &&&= \mathbb{P}(X \leq \ln y) \\ &&&= F(\ln y) \\ \Rightarrow && f_Y(y) &= f(\ln y)/y \\ \\ && f'_Y(y) &= \frac{f'(\ln y) - f(\ln y)}{y^2} \end{align*} Therefore since the mode satisfies \(f'_Y = 0\) we must have \(f'(\ln \lambda ) = f(\ln \lambda)\)
  3. This is the integral of the pdf of \(N(\mu + \sigma^2, \sigma^2)\) and therefore is clearly \(1\). \begin{align*} && \E[Y] &= \int_{-\infty}^{\infty} e^x \cdot \frac{1}{\sqrt{2\pi \sigma^2}} e^{-(x-\mu)^2/(2\sigma^2)} \d x \\ &&&= \frac{1}{\sqrt{2\pi \sigma^2}} \int_{-\infty}^{\infty} \exp (x - (x-\mu)^2/(2\sigma^2)) \d x\\ &&&= \frac{1}{\sqrt{2\pi \sigma^2}} \int_{-\infty}^{\infty} \exp ((2x \sigma^2- (x-\mu)^2)/(2\sigma^2)) \d x\\ &&&= \frac{1}{\sqrt{2\pi \sigma^2}} \int_{-\infty}^{\infty} \exp (-(x-\mu-\sigma^2)^2+2\mu \sigma^2-\sigma^4)/(2\sigma^2)) \d x\\ &&&= \frac{1}{\sqrt{2\pi \sigma^2}} \int_{-\infty}^{\infty} \exp (-(x-\mu+\sigma^2)^2)/(2\sigma^2)+\mu +\frac12\sigma^2) \d x\\ &&&= \e^{\mu +\frac12\sigma^2}\frac{1}{\sqrt{2\pi \sigma^2}} \int_{-\infty}^{\infty} \exp (-(x-\mu-\sigma^2)^2)/(2\sigma^2)) \d x\\ &&&= \e^{\mu +\frac12\sigma^2} \end{align*}
  4. Notice that \(y_m = e^\mu < e^{\mu + \tfrac12 \sigma^2} = \E[Y]\), so it suffices to prove that \(\lambda < e^{\mu}\) Notice that \(f'(x) - f(x) = f(x)[-(x-\mu)/\sigma^2 - 1]\) and therefore \(\ln y - \mu = -\sigma^2\) so \(\lambda = e^{\mu - \sigma^2}\) which is clearly less than \(e^{\mu}\) as required.

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2014 Paper 3 Q13
D: 1700.0 B: 1500.0
probability probability generating functions PGF conditional probability geometric distribution expectation comparing coefficients

I play a game which has repeated rounds. Before the first round, my score is \(0\). Each round can have three outcomes:

  1. my score is unchanged and the game ends;
  2. my score is unchanged and I continue to the next round;
  3. my score is increased by one and I continue to the next round.
The probabilities of these outcomes are \(a\), \(b\) and \(c\), respectively (the same in each round), where \(a+b+c=1\) and \(abc\ne0\). The random variable \(N\) represents my score at the end of a randomly chosen game. Let \(G(t)\) be the probability generating function of \(N\).
  1. Suppose in the first round, the game ends. Show that the probability generating function conditional on this happening is 1.
  2. Suppose in the first round, the game continues to the next round with no change in score. Show that the probability generating function conditional on this happening is \(G(t)\).
  3. By comparing the coefficients of \(t^n\), show that $ G(t) = a + bG(t) + ctG(t)\,. $ Deduce that, for \(n\ge0\), \[ P(N=n) = \frac{ac^n}{(1-b)^{n+1}}\,. \]
  4. Show further that, for \(n\ge0\), \[ P(N=n) = \frac{\mu^n}{(1+\mu)^{n+1}}\,, \] where \(\mu=\E(N)\).

Solution:

  1. If the game ends in the first round then the score is exactly \(0\) and the pgf is \(1\cdot x^0 = 1\)
  2. If the game moves onto the next round with no change in the first round then it's as if nothing happened, therefore the pgf is the original pgf \(G(t)\)
  3. If the game moves into the next round with the score increased by one, then the pgf is \(tG(t)\) since all the scores are increased by \(1\). Therefore \begin{align*} && G(t) &= \E[t^N] \\ &&&= \E[\E[t^N | \text{first round}]] \\ &&&= a + bG(t) + ctG(t) \\ \Rightarrow && G(t)(1-b-ct) = a \\ \Rightarrow && G(t) &= \frac{a}{(1-b)-ct} \\ &&&= \frac{a}{(1-b)} \frac{1}{1- \left(\frac{c}{1-b}\right)t} \\ &&&= \sum_{n=0}^\infty \frac{a}{1-b} \frac{c^n}{(1-b)^n} t^n\\ &&&= \sum_{n=0}^{\infty} \frac{ac^n}{(1-b)^{n+1}}t^n \end{align*} Therefore by comparing coefficients, \(\mathbb{P}(N=n) = \frac{ac^n}{(1-b)^{n+1}}\)
  4. \(\,\) \begin{align*} && \E[N] &= G'(1) \\ &&&= \frac{ac}{((1-b)-c)^2} \\ &&&= \frac{ac}{a^2} = \frac{c}{a} \\ \\ && \frac{\mu^n}{(1+\mu)^{n+1}} &= \frac{c^na^{-n}}{(a+c)^{n+1}a^{-n-1}} \\ &&&= \frac{ac^n}{(a+c)^{n+1}}\\ &&&= \frac{ac^n}{(1-b)^{n+1}}\\ &&&= \mathbb{P}(N=n) \end{align*} as required

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2013 Paper 1 Q12
D: 1500.0 B: 1468.0
probability combinatorics random selection binomial coefficient Stirling's approximation counting arguments conditional probability

Each day, I have to take \(k\) different types of medicine, one tablet of each. The tablets are identical in appearance. When I go on holiday for \(n\) days, I put \(n\) tablets of each type in a container and on each day of the holiday I select \(k\) tablets at random from the container.

  1. In the case \(k=3\), show that the probability that I will select one tablet of each type on the first day of a three-day holiday is \(\frac9{28}\). Write down the probability that I will be left with one tablet of each type on the last day (irrespective of the tablets I select on the first day).
  2. In the case \(k=3\), find the probability that I will select one tablet of each type on the first day of an \(n\)-day holiday.
  3. In the case \(k=2\), find the probability that I will select one tablet of each type on each day of an \(n\)-day holiday, and use Stirling's approximation \[ n!\approx \sqrt{2n\pi} \left(\frac n\e\right)^n \] to show that this probability is approximately \(2^{-n} \sqrt{n\pi\;}\).

Solution:

  1. The probability the first is different to the second is \(\frac68\), the probability the third is different to both of the first two is \(\frac37\) therefore the probability is \(\frac{6}{8} \cdot \frac37 = \frac9{28}\) Whatever pills we are left with on the last day is essentially the same random choice as we make on the first day, therefore \(\frac9{28}\)
  2. The probability the first is different to the second is \(\frac{2n}{3n-1}\), the probability the third is different to both of the first two is \(\frac{n}{3n-2}\) therefore the probability is \(\frac{2n^2}{(3n-1)(3n-2)}\). [We can also view this as \(\frac{(3n) \cdot (2n) \cdot n}{(3n) \cdot (3n-1) \cdot (3n-2)}\)]
  3. Suppose describe the pills as \(B\) and \(R\) and also number them, then we must have a sequence of the form: \[ B_1R_1 \, B_2R_2 \, B_3R_3 \, \cdots \, B_{n}R_n \] However, we can also rearrange the order of the \(B\) and \(R\) pills in \(n!\) ways each, and also the order of the pairs in \(2^n\) ways. There are \((2n)!\) orders we could have taken the pills out therefore the probability is \begin{align*} && P &= \frac{2^n (n!)^2}{(2n)!} = \frac{2^n}{\binom{2n}{n}} \\ &&&\approx \frac{2^n \cdot 2n \pi \left ( \frac{n}{e} \right)^{2n}}{\sqrt{2 \cdot 2n \cdot \pi} \left ( \frac{2n}{e} \right)^{2n}} \\ &&&= \frac{2^n \sqrt{n \pi} \cdot n^{2n} \cdot e^{-2n}}{2^{2n} \cdot n^{2n} \cdot e^{-2n}} \\ &&&= 2^{-n} \sqrt{n \pi} \end{align*} There is a nice way to think about this question using conditional probability. Suppose we are drawing out of an infinitely supply of \(R\) and \(B\) pills, then each day there is a \(\frac12\) chance of getting different pills. Therefore over \(n\) days there is a \(2^{-n}\) chance of getting different pills. Conditional on the balanced total we see that \begin{align*} && \mathbb{P}(\text{balanced every day} |\text{balanced total}) &= \frac{\mathbb{P}(\text{balanced every day})}{\mathbb{P}(\text{balanced total})} \end{align*} We have already seen the term that is balanced total is \(\frac{1}{2^{2n}}\binom{2n}{n}\), but we can also approximate the balanced total using a normal approximation. \(B(2n, \tfrac12) \approx N(n, \frac{n}{2})\) and so: \begin{align*} \mathbb{P}(X = n) &\approx \mathbb{P}\left (n-0.5 \leq \sqrt{\tfrac{n}{2}} Z + n \leq n+0.5 \right) \\ &= \mathbb{P}\left (- \frac1{\sqrt{2n}} \leq Z \leq \frac{1}{\sqrt{2n}} \right) \\ &= \int_{- \frac1{\sqrt{2n}}}^{\frac1{\sqrt{2n}}} \frac{1}{\sqrt{2\pi}} e^{-x^2/2} \d x \approx \frac{2}{\sqrt{2n}} \frac{1}{\sqrt{2\pi}} \\ &\approx \frac{1}{\sqrt{n\pi}} \end{align*}

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2013 Paper 1 Q13
D: 1516.0 B: 1532.0
probability discrete probability distribution expectation combinatorics hypergeometric complementary pairs random selection

From the integers \(1, 2, \ldots , 52\), I choose seven (distinct) integers at random, all choices being equally likely. From these seven, I discard any pair that sum to 53. Let \(X\) be the random variable the value of which is the number of discarded pairs. Find the probability distribution of \(X\) and show that \(\E (X) = \frac 7 {17}\). Note: \(7\times 17 \times 47 =5593\).

Solution: There are \(\binom{26}3\binom{23}{1}2\) ways to obtain \(3\) pairs There are \(\binom{26}2 \binom{24}3 \cdot 2^3\) ways to obtain \(2\) pairs There are \(\binom{26}1 \binom{25}5 \cdot 2^5\) ways to obtain \(1\) pairs There are \(\binom{26}7 \cdot 2^7\) ways to obtain \(0\) pairs There are \(\binom{52}{7}\) ways to choose our integers, so \begin{align*} && \mathbb{P}(X = 3) &= \frac{\binom{26}{3} \cdot \binom{23}{1} \cdot 2}{\binom{52}{7}} \\ &&&= \frac{7! \cdot 26 \cdot 25 \cdot 24 \cdot 23 \cdot 2}{3! \cdot 52 \cdot 51 \cdot 50 \cdot 49 \cdot 48 \cdot 47 \cdot 46} \\ &&&= \frac{7 \cdot 6 \cdot 5 \cdot 4 }{51 \cdot 2\cdot 49 \cdot 2\cdot 47 \cdot 2} \\ &&&= \frac{ 5 }{17\cdot 7\cdot 47} = \frac{5}{5593} \\ \\ && \mathbb{P}(X = 2) &= \frac{\binom{26}2 \binom{24}3 \cdot 2^3}{\binom{52}{7}} \\ &&&= \frac{220}{5593} \\ \\ && \mathbb{P}(X = 1) &= \frac{\binom{26}1 \binom{25}5 \cdot 2^5}{\binom{52}{7}} \\ &&&= \frac{1848}{5593} \\ \\ && \mathbb{P}(X = 0) &= \frac{\binom{26}7 \cdot 2^7}{\binom{52}{7}} \\ &&&= \frac{3520}{5593} \\ \\ && \mathbb{E}(X) &= \frac{1848}{5593} + 2 \cdot \frac{220}{5593} + 3 \cdot \frac{5}{5593} \\ &&&= \frac{2303}{5593} = \frac{7}{17} \end{align*} Notice we can find the expected value directly: Let \(X_i\) be the random variable the \(i\)th number is discarded. Notice that \(\mathbb{E}(X) = \mathbb{E}\left (\frac12 \left (X_1 +X_2 +X_3 +X_4 +X_5 +X_6 +X_7\right) \right)\) and also notice that each \(X_i\) has the same distribution (although not independent!). Then \begin{align*} &&\mathbb{E}(X) &= \frac72 \mathbb{E}(X_i) \\ &&&= \frac72 \cdot \left (1 - \frac{50}{51} \cdot \frac{49}{50} \cdots \frac{45}{46} \right) = \frac74 \left ( 1 - \frac{45}{51}\right) \\ &&&= \frac72 \cdot \frac{6}{51} \\ &&&= \frac7{17} \end{align*}

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2013 Paper 2 Q12
D: 1600.0 B: 1484.0
probability Poisson distribution expectation variance exponential series hyperbolic functions conditional distribution independence

The random variable \(U\) has a Poisson distribution with parameter \(\lambda\). The random variables \(X\) and \(Y\) are defined as follows. \begin{align*} X&= \begin{cases} U & \text{ if \(U\) is 1, 3, 5, 7, \(\ldots\,\)} \\ 0 & \text{ otherwise} \end{cases} \\ Y&= \begin{cases} U & \text{ if \(U\) is 2, 4, 6, 8, \(\ldots\,\) } \\ 0 & \text{ otherwise} \end{cases} \end{align*}

  1. Find \(\E(X)\) and \(\E(Y)\) in terms of \(\lambda\), \(\alpha\) and \(\beta\), where \[ \alpha = 1+\frac{\lambda^2}{2!}+\frac{\lambda^4}{4!} +\cdots\, \text{ \ \ and \ \ } \beta = \frac{\lambda}{1!} + \frac{\lambda^3}{3!} + \frac{\lambda^5}{5!} +\cdots\,. \]
  2. Show that \[ \var(X) = \frac{\lambda\alpha+\lambda^2\beta}{\alpha+\beta} - \frac{\lambda^2\alpha^2}{(\alpha+\beta)^2} \] and obtain the corresponding expression for \(\var(Y)\). Are there any non-zero values of \(\lambda\) for which \( \var(X) + \var(Y) = \var(X+Y)\,\)?

Solution:

  1. \begin{align*} \mathbb{E}(X) &= \sum_{r=1}^\infty r \mathbb{P}(X = r) \\ &= \sum_{j=1}^{\infty} (2j-1)\mathbb{P}(U=2j-1) \\ &= \sum_{j=1}^{\infty}(2j-1) \frac{e^{-\lambda} \lambda^{2j-1}}{(2j-1)!} \\ &= \sum_{j=1}^{\infty} e^{-\lambda} \frac{\lambda^{2j-1}}{(2j-2)!} \\ &= \lambda e^{-\lambda} \sum_{j=1}^{\infty} \frac{\lambda^{2j-2}}{(2j-2)!} \\ &= \lambda e^{-\lambda} \alpha \end{align*} Since \(\mathbb{E}(X+Y) = \lambda, \mathbb{E}(Y) = \lambda(1-e^{-\lambda}\alpha) = \lambda(e^{-\lambda}(\alpha+\beta) - e^{-\lambda}\alpha) = \lambda e^{-\lambda} \beta\). Alternatively, as \(\beta + \alpha = e^{\lambda}\), \(\mathbb{E}(X) = \frac{\lambda \alpha}{\alpha+\beta}, \mathbb{E}(Y) = \frac{\lambda \beta}{\alpha+\beta}\)
  2. \begin{align*} \textrm{Var}(X) &= \mathbb{E}(X^2) - [\mathbb{E}(X) ]^2 \\ &= \sum_{odd} r^2 \mathbb{P}(U = r) - \left [ \mathbb{E}(X) \right]^2 \\ &= \sum_{odd} (r(r-1)+r)\frac{e^{-\lambda}\lambda^r}{r!} - \frac{\lambda^2 \alpha^2}{(\alpha+\beta)^2} \\ &= \sum_{odd} \frac{e^{-\lambda}\lambda^r}{(r-2)!}+\sum_{odd} \frac{e^{-\lambda}\lambda^r}{(r-1)!} - \frac{\lambda^2 \alpha^2}{(\alpha+\beta)^2} \\ &= e^{-\lambda}\lambda^2 \beta + e^{-\lambda}\lambda \alpha - \frac{\lambda^2 \alpha^2}{(\alpha+\beta)^2} \\ &= \frac{\lambda \alpha + \lambda^2 \beta}{\alpha+\beta}- \frac{\lambda^2 \alpha^2}{(\alpha+\beta)^2} \end{align*} Similarly, \begin{align*} \textrm{Var}(Y) &= \mathbb{E}(Y^2) - [\mathbb{E}(Y) ]^2 \\ &= \sum_{even} r^2 \mathbb{P}(U = r) - \left [ \mathbb{E}(Y) \right]^2 \\ &= \sum_{even} (r(r-1)+r)\frac{e^{-\lambda}\lambda^r}{r!} - \frac{\lambda^2 \beta^2}{(\alpha+\beta)^2} \\ &= e^{-\lambda}\lambda^2\alpha + e^{-\lambda}\lambda \beta - \frac{\lambda^2 \beta^2}{(\alpha+\beta)^2} \\ &= \frac{\lambda \beta + \lambda^2 \alpha}{\alpha+\beta}- \frac{\lambda^2 \beta^2}{(\alpha+\beta)^2} \end{align*} Since \(\textrm{Var}(X+Y) = \textrm{Var}(U) = \lambda\), we are interested in solving: \begin{align*} \lambda &= \frac{\lambda \alpha + \lambda^2 \beta}{\alpha+\beta}- \frac{\lambda^2 \alpha^2}{(\alpha+\beta)^2} + \frac{\lambda \beta + \lambda^2 \alpha}{\alpha+\beta}- \frac{\lambda^2 \beta^2}{(\alpha+\beta)^2} \\ &= \frac{\lambda(\alpha+\beta) + \lambda^2(\alpha+\beta)}{\alpha+\beta} - \frac{\lambda^2(\alpha^2+\beta^2)}{(\alpha+\beta)^2} \\ &= \lambda + \lambda^2 \frac{(\alpha+\beta)^2 - (\alpha^2+\beta^2)}{(\alpha+\beta)^2} \\ &= \lambda + \lambda^2 \frac{2\alpha\beta}{(\alpha+\beta)^2} \end{align*} which is clearly not possible if \(\lambda \neq 0\)

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2013 Paper 2 Q13
D: 1600.0 B: 1516.0
probability geometric distribution biased coin runs inequality sequences combinatorial probability

A biased coin has probability \(p\) of showing a head and probability \(q\) of showing a tail, where \(p\ne0\), \(q\ne0\) and \(p\ne q\). When the coin is tossed repeatedly, runs occur. A straight run of length \(n\) is a sequence of \(n\) consecutive heads or \(n\) consecutive tails. An alternating run of length \(n\) is a sequence of length \(n\) alternating between heads and tails. An alternating run can start with either a head or a tail. Let \(S\) be the length of the longest straight run beginning with the first toss and let \(A\) be the length of the longest alternating run beginning with the first toss.

  1. Explain why \(\P(A=1)=p^2+q^2\) and find \(\P(S=1)\). Show that \(\P(S=1)<\P(A=1)\).
  2. Show that \(\P(S=2)= \P(A=2)\) and determine the relationship between \(\P(S=3)\) and \( \P(A=3)\).
  3. Show that, for \(n>1\), \(\P(S=2n)>\P(A=2n)\) and determine the corresponding relationship between \(\P(S=2n+1)\) and \(\P(A=2n+1)\). [You are advised not to use \(p+q=1\) in this part.]

Solution:

  1. The only way \(A = 1\) is if we get \(HH\) or \(TT\) which has probability \(p^2+q^2\). The only way we get \(S=1\) is if we have \(HT\) to \(TH\), ie \(2pq\). Since \((p-q)^2 = p^2 + q^2 - 2pq >0\) we must have \(\mathbb{P}(A=1) > \mathbb{P}(S=1)\).
  2. \(\,\) \begin{align*} \mathbb{P}(S=2) &= p^2q + q^2p \\ \mathbb{P}(A=2) &= pq^2 + qp^2 = \mathbb{P}(S=2) \\ \\ \mathbb{P}(S=3) &= p^3q + q^3p = pq(p^2+q^2) \\ \mathbb{P}(A=3) &= pqp^2 + qpq^2 = pq(p^2+q^2) = \mathbb{P}(S=3) \end{align*}
  3. For \(n > 1\) we must have \begin{align*} && \mathbb{P}(S = 2n) &= p^{2n}q + q^{2n}p \\ && \mathbb{P}(A=2n) &= (pq)^{n}q + (qp)^{n}p \\ &&&= p^nq^{n+1} + q^np^{n+1} \\ && \mathbb{P}(S = 2n) &> \mathbb{P}(A = 2n) \\ \Leftrightarrow && p^{2n}q + q^{2n}p & > p^nq^{n+1} + q^np^{n+1}\\ \Leftrightarrow && 0 & < p^{2n}q+q^{2n}p - p^nq^{n+1} -q^np^{n+1}\\ &&&= (p^n-q^n)(qp^n - pq^n) \end{align*} which is clearly true. \begin{align*} && \mathbb{P}(S=2n+1) &= p^{2n+1}q + q^{2n+1}p \\ && \mathbb{P}(A=2n+1) &= (pq)^{n}p^2 + (qp)^{n}q^2 \\ &&&= p^{n+2}q^n + q^{n+2}p^n \end{align*} The same factoring logic shows that \(\mathbb{P}(S = 2n+1) > \mathbb{P}(A=2n+1)\)

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2013 Paper 3 Q12
D: 1700.0 B: 1500.0
probability expectation variance indicator random variables combinatorics bivariate data covariance summation

A list consists only of letters \(A\) and \(B\) arranged in a row. In the list, there are \(a\) letter \(A\)s and \(b\) letter \(B\)s, where \(a\ge2\) and \(b\ge2\), and \(a+b=n\). Each possible ordering of the letters is equally probable. The random variable \(X_1\) is defined by \[ X_1 = \begin{cases} 1 & \text{if the first letter in the row is \(A\)};\\ 0 & \text{otherwise.} \end{cases} \] The random variables \(X_k\) (\(2 \le k \le n\)) are defined by \[ X_k = \begin{cases} 1 & \text{if the \((k-1)\)th letter is \(B\) and the \(k\)th is \(A\)};\\ 0 & \text{otherwise.} \end{cases} \] The random variable \(S\) is defined by \(S = \sum\limits_ {i=1}^n X_i\,\).

  1. Find expressions for \(\E(X_i)\), distinguishing between the cases \(i=1\) and \(i\ne1\), and show that \(\E(S)= \dfrac{a(b+1)}n\,\).
  2. Show that:
    1. for \(j\ge3\), \(\E(X_1X_j) = \dfrac{a(a-1)b}{n(n-1)(n-2)}\,\);
    2. \[ \sum\limits_{i=2}^{n-2} \bigg( \sum\limits_{j=i+2}^n \E(X_iX_j)\bigg) = \dfrac{a(a-1)b(b-1)}{2n(n-1)}\,\]
    3. \(\var(S) = \dfrac {a(a-1)b(b+1)}{n^2(n-1)}\,\).

Solution:

  1. Notice that \(\E[X_1] = \frac{a}{n}\) and consider \(\E[X_i]\) with \(i > 1\). the probability that this is \(1\) is \(\frac{b}{n} \cdot \frac{a}{n-1}\). So \begin{align*} && \E[S] &= \E[X_1] + \sum_{i=2}^n \E[X_i] \\ &&&= \frac{a}{n} + (n-1) \frac{ab}{n(n-1)} \\ &&&= \frac{a(b+1)}{n} \end{align*}
    1. The probability \(X_1X_j = 1\) is \(\frac{a}{n} \cdot \frac{b}{n-1} \cdot \frac{a-1}{n-2} = \frac{a(a-1)b}{n(n-1)(n-2)}\) since there is nothing special about the order, and the first is an \(A\) with probability \(\frac{a}{n}\) and given this occurs there are now \(a-1\) \(A\) and \(n-1\) letters left etc... Therefore \(\E[X_1X_j] = \frac{a(a-1)b}{n(n-1)(n-2)}\)
    2. \(\E[X_iX_j]\) when the pairs don't overlap is \(\frac{a}{n} \frac{b}{n-1} \frac{a-1}{n-2} \frac{b-1}{n-3}\), and so \begin{align*} && \sum\limits_{i=2}^{n-2} \bigg( \sum\limits_{j=i+2}^n \E(X_iX_j)\bigg) &= \sum\limits_{i=2}^{n-2} \bigg( \sum\limits_{j=i+2}^n \frac{a(a-1)b(b-1)}{n(n-1)(n-2)(n-3)}\bigg) \\ &&&= \frac{a(a-1)b(b-1)}{n(n-1)(n-2)(n-3)}\sum\limits_{i=2}^{n-2} \bigg( \sum\limits_{j=i+2}^n 1\bigg) \\ &&&= \frac{a(a-1)b(b-1)}{n(n-1)(n-2)(n-3)}\sum\limits_{i=2}^{n-2} (n-(i+1)) \\ &&&= \frac{a(a-1)b(b-1)}{n(n-1)(n-2)(n-3)} \left ((n-1)(n-3)-\frac{(n-2)(n-1)}{2}+1 \right) \\ &&&= \frac{a(a-1)b(b-1)}{n(n-1)(n-2)(n-3)} \left ( \frac{2n^2-8n-6-n^2+3n-2+2}{2}\right) \\ &&&= \frac{a(a-1)b(b-1)}{n(n-1)(n-2)(n-3)} \left ( \frac{n^2-5n-6}{2}\right) \\ &&&= \frac{a(a-1)b(b-1)}{2n(n-1)} \end{align*}
    3. We also need to consider the other cross terms. \(X_iX_{i+1}=0\). (Since \(X_i = 1\) means the \(i\)th letter is \(A\) and \(X_{i+1} = 1\) means the \(i\)th letter is \(B\)). It's the same story for \(X_1X_2\), and so all the cross terms are accounted for. Therefore \begin{align*} && \E[S^2] &= \E \left [\sum X_i^2 + 2\sum_{i \neq j} X_i X_j \right] \\ &&&= \frac{a(b+1)}{n} +2(n-2)\frac{a(a-1)b}{n(n-1)(n-2)}+ 2 \frac{a(a-1)b(b-1)}{2n(n-1)} \\ &&&= \frac{a(b+1)}{n} +\frac{2a(a-1)b}{n(n-1)} + \frac{a(a-1)b(b-1)}{n(n-1)} \\ &&&= \frac{a(b+1)}{n} +\frac{a(a-1)b(b+1)}{n(n-1)} \\ && \var[S] &= \E[S^2] - \left ( \E[S] \right)^2 \\ &&&= \frac{a(b+1)}{n} + \frac{a(a-1)b(b+1)}{n(n-1)} - \frac{a^2(b+1)^2}{n^2} \\ &&&= \frac{a(b+1) \left (n(n-1) + (a-1)b n -a(b+1)(n-1) \right)}{n^2(n-1)} \\ &&&= \frac{a(b+1) \left ( (n-a)(n-b-1) \right)}{n^2(n-1)} \\ &&&= \frac{a(b+1) \left ( b(a-1) \right)}{n^2(n-1)} \\ \end{align*}

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2012 Paper 1 Q12
D: 1484.0 B: 1516.0
probability conditional probability Bayes' theorem continuous probability distribution PDF integration fire extinguisher testing law of total probability

Fire extinguishers may become faulty at any time after manufacture and are tested annually on the anniversary of manufacture. The time \(T\) years after manufacture until a fire extinguisher becomes faulty is modelled by the continuous probability density function \[ f(t) = \begin{cases} \frac{2t}{(1+t^2)^2}& \text{for \(t\ge0\)}\,,\\[4mm] \ \ \ \ 0& \text{otherwise}. \end{cases} \] A faulty fire extinguisher will fail an annual test with probability \(p\), in which case it is destroyed immediately. A non-faulty fire extinguisher will always pass the test. All of the annual tests are independent. Show that the probability that a randomly chosen fire extinguisher will be destroyed exactly three years after its manufacture is \(p(5p^2-13p +9)/10\). Find the probability that a randomly chosen fire extinguisher that was destroyed exactly three years after its manufacture was faulty 18 months after its manufacture.

Solution: The probability it becomes faulty in each year is: \begin{align*} \mathbb{P}(\text{faulty in Y}1) &= \int_0^1 \frac{2t}{(1+t^2)^2} \, dt \\ &= \left [ -\frac{1}{(1+t^2)} \right]_0^1 \\ &= 1 - \frac{1}{2} = \frac{1}{2} \\ \mathbb{P}(\text{faulty in Y}2) &= \frac{1}{2} - \frac{1}{5} = \frac{3}{10} \\ \mathbb{P}(\text{faulty in Y}3) &= \frac{1}{5} - \frac{1}{10} = \frac{1}{10} \end{align*} The probability of failing for the first time after exactly \(3\) years is: \begin{align*} \mathbb{P}(\text{faulty in Y1, }PPF) &+ \mathbb{P}(\text{faulty in Y2, }PF) + + \mathbb{P}(\text{faulty in Y3, }F) \\ &= \frac12 (1-p)^2p + \frac3{10}(1-p)p + \frac1{10}p \\ &= \frac{p}{10} \l 5(1-p)^2 + 3(1-p) + 1 \r \\ &= \frac{p}{10} \l 5 - 10p + 5p^2 + 3 -3p +1 \r \\ &= \frac{p}{10} \l 9 - 13p + 5p^2 \r \end{align*} as required. The probability that a randomly chosen fire extinguisher that was destroyed exactly three years after its manufacture was faulty 18 months after its manufacture is: \begin{align*} \mathbb{P}(\text{faulty 18 months after} | \text{fails after 3 tries}) &= \frac{\mathbb{P}(\text{faulty 18 months after and fails after 3 tries})}{\mathbb{P}(\text{fails after exactly 3 tries})} \end{align*} We can compute \(\mathbb{P}(\text{faulty 18 months after and fails after 3 tries})\) by looking at \(2\) cases, fails between \(12\) months and \(18\) years, and between \(0\) years and \(1\) year. \begin{align*} \mathbb{P}(\text{faulty between 1y and 18m}) &= \int_{1}^{\frac32} \frac{2t}{(1+t^2)^2} \, dt \\ &= \left [ -\frac{1}{(1+t^2)} \right]_{1}^{\frac32} \\ &= \frac12 - \frac{4}{13} = \frac{5}{26} \\ \end{align*} So the probability is: \begin{align*} \mathbb{P} &= \frac{\frac{5}{26}(1-p)p + \frac12(1-p)^2p}{\frac{p}{10} \l 9 - 13p + 5p^2 \r} \\ &= \frac{\frac{25}{13}(1-p) + 5(1-p)^2}{9 - 13p + 5p^2} \\ &= \frac{5}{13} \frac{(1-p)\l 5 + 13(1-p) \r}{9 - 13p + 5p^2} \\ &= \frac{5}{13} \frac{(1-p)\l 18 - 13p \r}{9 - 13p + 5p^2} \\ \end{align*}

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2012 Paper 1 Q13
D: 1500.0 B: 1529.2
probability conditional expected value combinatorics counting digits inclusion-exclusion

I choose at random an integer in the range 10000 to 99999, all choices being equally likely. Given that my choice does not contain the digits 0, 6, 7, 8 or 9, show that the expected number of different digits in my choice is 3.3616.

Solution: We are choosing any \(5\) digit number from \(\{1,2,3,4,5\}\). There are \(5^5\) such numbers. \begin{align*} && \mathbb{E}(\text{different digits}) &= \frac1{5^5} \left (1 \cdot 5 + 2 \cdot \binom{5}{2}(2^5-2)+3 \cdot \binom{5}{3}(3^5-3 \cdot 2^5+3)+4 \cdot \binom{5}{4}(4^5 - 4 \cdot 3^5+6 \cdot 2^5-4) + 5 \cdot 5! \right) \\ &&&= \frac{2101}{625} = 3.3616 \end{align*}

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2012 Paper 2 Q12
D: 1600.0 B: 1500.7
probability conditional Bayes' theorem tree diagram binomial distribution mode inequality optimisation

A modern villa has complicated lighting controls. In order for the light in the swimming pool to be on, a particular switch in the hallway must be on and a particular switch in the kitchen must be on. There are four identical switches in the hallway and four identical switches in the kitchen. Guests cannot tell whether the switches are on or off, or what they control. Each Monday morning a guest arrives, and the switches in the hallway are either all on or all off. The probability that they are all on is \(p\) and the probability that they are all off is \(1-p\). The switches in the kitchen are each on or off, independently, with probability \(\frac12\).

  1. On the first Monday, a guest presses one switch in the hallway at random and one switch in the kitchen at random. Find the probability that the swimming pool light is on at the end of this process. Show that the probability that the guest has pressed the swimming pool light switch in the hallway, given that the light is on at the end of the process, is \(\displaystyle \frac{1-p}{1+2p}\).
  2. On each of seven Mondays, guests go through the above process independently of each other, and each time the swimming pool light is found to be on at the end of the process. Given that the most likely number of days on which the swimming pool light switch in the hallway was pressed is 3, show that \(\frac14 < p < \frac{5}{14}\).

Solution:

  1. \(\,\) \begin{align*} && \mathbb{P}(\text{hall switch on}) &= \underbrace{p \cdot \frac34 }_{\text{already on and not flipped}}+ \underbrace{(1-p) \cdot \frac14}_{\text{not on and flipped}} \\ &&&= \frac14 +\frac12 p\\ && \mathbb{P}(\text{kitchen on}) &= \frac12 \\ \Rightarrow && \mathbb{P}(\text{pool is on}) &= \frac18 + \frac14p \end{align*} \begin{align*} && \mathbb{P}(\text{flipped hall switch} | \text{pool on}) &= \frac{\mathbb{P}(\text{flipped hall and pool on})}{\mathbb{P}(\text{pool on})} \\ &&&= \frac{(1-p)\frac14 \cdot \frac 12}{\frac18 + \frac14 p} \\ &&&= \frac{1-p}{1+2p} \end{align*}
  2. The number of days the swimming pool light was pressed is \(X = B\left (7, \frac{1-p}{1+2p} \right)\), and we have that \(\mathbb{P}(X = 2) < \mathbb{P}(X = 3) > \mathbb{P}(X=4)\) (since the binomial is unimodal). Let \(q = \frac{1-p}{1+2p} \) \begin{align*} && \mathbb{P}(X = 2) &< \mathbb{P}(X = 3) \\ \Rightarrow && \binom{7}{2} q^2(1-q)^5 &< \binom{7}{3}q^3(1-q)^4 \\ \Rightarrow && 21(1-q) &< 35q \\ \Rightarrow && 21 &< 56q \\ \Rightarrow && \frac{3}{8} &< \frac{1-p}{1+2p} \\ \Rightarrow && 3+6p &< 8-8p \\ \Rightarrow && 14p &< 5\\ \Rightarrow && p &< \frac5{14} \\ \\ && \mathbb{P}(X = 3) &> \mathbb{P}(X = 4) \\ \Rightarrow && \binom{7}{3} q^3(1-q)^4 &> \binom{7}{4}q^4(1-q)^3 \\ \Rightarrow &&(1-q)&> q \\ \Rightarrow && \frac12 &> q \\ \Rightarrow && \frac12 &> \frac{1-p}{1+2p} \\ \Rightarrow && 1+2p &> 2-2p \\ \Rightarrow && 4p &> 1\\ \Rightarrow && p &> \frac1{4} \end{align*} Therefore \(\frac14 < p < \frac{5}{14}\) as required.

View
2011 Paper 1 Q12
D: 1500.0 B: 1470.2
probability conditional probability combinatorics counting arrangements ballot problem queuing change-making case analysis

I am selling raffle tickets for \(\pounds1\) per ticket. In the queue for tickets, there are \(m\) people each with a single \(\pounds1\) coin and \(n\) people each with a single \(\pounds2\) coin. Each person in the queue wants to buy a single raffle ticket and each arrangement of people in the queue is equally likely to occur. Initially, I have no coins and a large supply of tickets. I stop selling tickets if I cannot give the required change.

  1. In the case \(n=1\) and \(m\ge1\), find the probability that I am able to sell one ticket to each person in the queue.
  2. By considering the first three people in the queue, show that the probability that I am able to sell one ticket to each person in the queue in the case \(n=2\) and \(m\ge2\) is \(\dfrac{m-1}{m+1}\,\).
  3. Show that the probability that I am able to sell one ticket to each person in the queue in the case \(n=3\) and \(m\ge3\) is \(\dfrac{m-2}{m+1}\,\).

Solution:

  1. The only way you wont be able to sell to them is if they are first, ie \(\frac1{m+1}\)
  2. If \(n=2\), the the only way you fail to sell to them is if one comes first or they both appear before two people with pound coins, ie \(2\) or \(122\). These have probabilities \(\frac{2}{m+2}\) and \(\frac{m}{m+2} \cdot \frac{2}{m+1} \frac{1}{m} = \frac{2}{(m+1)(m+2)}\). Therefore the total probability you don't sell all the tickets is \(\frac{2}{m+2}\left ( 1 + \frac{1}{m+1} \right) = \frac{2}{m+2} \frac{m+2}{m+1} = \frac{2}{m+1}\). Therefore the probability you do sell all the tickets is \(1 - \frac{2}{m+1} = \frac{m-1}{m+1}\)
  3. The only ways to fail when \(n=3\) are: \(2\), \(122\), or if all three \(2\)s appear before three \(1\)s. this can happen in \(11222\), \(12122\) These happen with probability: \begin{align*} 2: && \frac{3}{m+3} \\ 122: && \frac{m}{m+3} \cdot \frac{3}{m+2} \cdot \frac{2}{m+1} \\ 11222: && \frac{m(m-1) 6}{(m+3)(m+2)(m+1)m(m-1)} \\ 12122: && \frac{m(m-1) 6}{(m+3)(m+2)(m+1)m(m-1)} \\ \end{align*} Therefore the total probability is: \begin{align*} P &= \frac{1}{(m+3)(m+2)(m+1)} \left (3(m+2)(m+1)+6m + 12 \right) \\ &= \frac{1}{(m+3)(m+2)(m+1)} \left (3(m+1)(m+2) \right) \\ &= \frac{3}{m+1} \end{align*} and the result follows

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2011 Paper 1 Q13
D: 1484.0 B: 1471.5
probability continuous probability distribution probability density function integration mean quantile piecewise function arcsin

In this question, you may use without proof the following result: \[ \int \sqrt{4-x^2}\, \d x = 2 \arcsin (\tfrac12 x ) + \tfrac 12 x \sqrt{4-x^2} +c\,. \] A random variable \(X\) has probability density function \(\f\) given by \[ \f(x) = \begin{cases} 2k & -a\le x <0 \\[3mm] k\sqrt{4-x^2} & \phantom{-} 0\le x \le 2 \\[3mm] 0 & \phantom{-}\text{otherwise}, \end{cases} \] where \(k\) and \(a\) are positive constants.

  1. Find, in terms of \(a\), the mean of \(X\).
  2. Let \(d\) be the value of \(X\) such that \(\P(X> d)=\frac1 {10}\,\). Show that \(d < 0\) if \(2a> 9\pi\) and find an expression for \(d\) in terms of \(a\) in this case.
  3. Given that \(d=\sqrt 2\), find \(a\).

Solution: First notice that \begin{align*} && 1 &= \int_{-a}^2 f(x) \d x \\ &&&= 2ka + k\pi \\ \Rightarrow && k &= (\pi + 2a)^{-1} \end{align*}

  1. \(\,\) \begin{align*} && \E[X] &= \int_{-a}^2 x f(x) \d x\\ &&&= \int_{-a}^0 2kx \d x + k\int_0^{2} x\sqrt{4-x^2} \d x\\ &&&= \left [kx^2 \right]_{-a}^0 +k \left [-\frac13(4-x^2)^{\frac32} \right]_0^2 \\ &&&= -ka^2 + \frac83k \\ &&&= \frac{\frac83-a^2}{\pi + 2a} \end{align*}
  2. Consider \(\mathbb{P}(X < 0)\) then \(d < 0 \Leftrightarrow \mathbb{P}(X < 0) > \frac{9}{10}\) \begin{align*} && \frac{9}{10} &< \mathbb{P}(X < 0) \\ &&&= \int_{-a}^0 2k \d x \\ &&&= \frac{2a}{\pi+2a} \\ \Leftrightarrow && 9\pi &< 2a \\ \\ && \frac{9}{10} &= \int_{-a}^d 2k \d x \\ &&&= \frac{2(d+a)}{\pi + 2a} \\ \Rightarrow && 9\pi &= 2a + 20d \\ \Rightarrow && d &= \frac{2a-9\pi}{20} \end{align*}
  3. Suppose \(d=\sqrt 2\) then \begin{align*} && \frac1{10} &= \int_{\sqrt{2}}^2 f(x) \d x \\ &&&= \int_{\sqrt{2}}^2 k\sqrt{4-x^2} \d x \\ &&&= k\left [ 2 \sin^{-1} \tfrac12 x + \tfrac12 x \sqrt{4-x^2}\right]_{\sqrt{2}}^2 \\ &&&= k\left (\pi -\frac{\pi}{2} - 1 \right) \\ \Rightarrow && \pi + 2a &= 5\pi - 10 \\ \Rightarrow && a &= 2\pi-5 \end{align*}

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2011 Paper 2 Q12
D: 1600.0 B: 1484.0
probability tree diagram conditional probability geometric series fair game expected value quadratic inequality match/game structure

Xavier and Younis are playing a match. The match consists of a series of games and each game consists of three points. Xavier has probability \(p\) and Younis has probability \(1-p\) of winning the first point of any game. In the second and third points of each game, the player who won the previous point has probability \(p\) and the player who lost the previous point has probability \(1-p\) of winning the point. If a player wins two consecutive points in a single game, the match ends and that player has won; otherwise the match continues with another game.

  1. Let \(w\) be the probability that Younis wins the match. Show that, for \(p\ne0\), \[ w = \frac{1-p^2}{2-p}. \] Show that \(w>\frac12\) if \(p<\frac12\), and \(w<\frac12\) if \(p>\frac12\). Does \(w\) increase whenever \(p\) decreases?
  2. If Xavier wins the match, Younis gives him \(\pounds1\); if Younis wins the match, Xavier gives him \(\pounds k\). Find the value of \(k\) for which the game is `fair' in the case when \(p =\frac23\).
  3. What happens when \(p = 0\)?

Solution:

  1. We can be in several states.
    1. No points played
    2. Y just won the last point
    3. X just won the last point
    4. Y won the game
    5. X won the game
    The probability \(Y\) wins from any of these states are: \begin{align*} &&P_{-} &= p P_X + (1-p) P_Y &= w \\ &&P_Y &= p + (1-p)P_X \\ &&P_X &= (1-p)P_Y \\ \\ \Rightarrow &&& \begin{cases} P_Y - (1-p)P_X &= p \\ (1-p)P_Y -P_X &= 0\end{cases} \\ \Rightarrow && P_Y &= \frac{1}{1-(1-p)^2} \cdot p \\ &&&= \frac{1}{2-p} \\ && P_X &= \frac{1-p}{2-p} \\ && w &= \frac{p(1-p) + (1-p)}{2-p} \\ &&&= \frac{1-p^2}{2-p} \end{align*}
  2. If \(p = \frac23\) then \(w = \frac{1-\frac49}{2-\frac23} = \frac{5}{12}\). The game is fair if \(\mathbb{E}(result) = 0\), ie \(\frac{5}{12} \cdot k - \frac{7}{12} 1 \Rightarrow k = \frac{7}{5} = 1.4\)
  3. If \(p = 0\) then they will keep playing forever, since no-one can win two points in a row.

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2011 Paper 3 Q13
D: 1700.0 B: 1500.0
probability binomial distribution hypergeometric distribution mode of distribution ratio of probabilities floor function sampling with replacement sampling without replacement

In this question, the notation \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \(x\), so for example \(\lfloor \pi\rfloor = 3\) and \(\lfloor 3 \rfloor =3\).

  1. A bag contains \(n\) balls, of which \(b\) are black. A sample of \(k\) balls is drawn, one after another, at random with replacement. The random variable \(X\) denotes the number of black balls in the sample. By considering \[ \frac{\P(X=r+1)}{\P(X=r)}\,, \] show that, in the case that it is unique, the most probable number of black balls in the sample is \[ \left\lfloor \frac{(k+1)b}{n}\right\rfloor. \] Under what circumstances is the answer not unique?
  2. A bag contains \(n\) balls, of which \(b\) are black. A sample of \(k\) balls (where \(k\le b\)) is drawn, one after another, at random without replacement. Find, in the case that it is unique, the most probable number of black balls in the sample. Under what circumstances is the answer not unique?

Solution:

  1. \(\mathbb{P}(X = r) = \binom{k}{r}p^rq^{k-r}\) where \(p = \frac{b}{n}, q = 1-p\). Therefore \begin{align*} && \frac{\mathbb{P}(X=r+1)}{\mathbb{P}(X=r)} &= \frac{\binom{k}{r+1}p^{r+1}q^{k-r-1}}{\binom{k}{r}p^rq^{k-r}} \\ &&&= \frac{(k-r)p}{(r+1)q} \\ &&&= \frac{(k-r)b}{(r+1)(n-b)} \end{align*} Comparing this to \(1\) we find: \begin{align*} && 1 & < \frac{\mathbb{P}(X=r+1)}{\mathbb{P}(X=r)} \\ \Leftrightarrow && 1 &< \frac{(k-r)b}{(r+1)(n-b)} \\ \Leftrightarrow && (r+1)(n-b) &<(k-r)b \\ \Leftrightarrow && rn& < (k+1)b - n \\ \Leftrightarrow && r &< \frac{(k+1)b}{n} - 1\\ \end{align*} If this equation is true, then \(\mathbb{P}(X=r+1)\) is larger, so \(r_{max} = \left \lfloor \frac{(k+1)b}{n} \right \rfloor\)
  2. Let \(Y\) be the number of black balls in our sample, ie \(\mathbb{P}(Y = r) = \binom{b}{r}\binom{n-b}{k-r}/\binom{n}{k}\), so \begin{align*} && \frac{\mathbb{P}(Y = r+1)}{\mathbb{P}(Y=r)} &= \frac{\binom{b}{r+1}\binom{n-b}{k-(r+1)}/\binom{n}{k}}{\binom{b}{r}\binom{n-b}{k-r}/\binom{n}{k}} \\ &&&= \frac{b-r}{r+1} \frac{k-r}{n-b-k+r+1} \\ && 1 &< \frac{\mathbb{P}(Y = r+1)}{\mathbb{P}(Y=r)} \\ \Leftrightarrow && (r+1)(n-b-k+r+1) &< (b-r)(k-r) \\ \Leftrightarrow &&r(n-b-k+1)+(n-b-k+1) &< -r(b+k)+bk \\ \Leftrightarrow &&r(n+1) &< bk+b+k+1-n \\ \Leftrightarrow && r &< \frac{(b+1)(k+1)}{n+1} - \frac{n}{n+1} \end{align*} Therefore \(r = \left \lfloor \frac{ (b+1)(k+1)}{n+1}\right \rfloor\), it is not unique if \(n+1\) divides \((b+1)(k+1)\)

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