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2017 Paper 1 Q12
D: 1500.0 B: 1513.9
probability expected value uniform distribution lottery binomial distribution approximation exponential approximation optimisation

In a lottery, each of the \(N\) participants pays \(\pounds c\) to the organiser and picks a number from \(1\) to \(N\). The organiser picks at random the winning number from \(1\) to \(N\) and all those participants who picked this number receive an equal share of the prize, \(\pounds J\).

  1. The participants pick their numbers independently and with equal probability. Obtain an expression for the probability that no participant picks the winning number, and hence determine the organiser's expected profit. Use the approximation \[ \left( 1 - \frac{a}{N} \right)^N \approx \e^{-a} \tag{\(*\)} \] to show that if \(2Nc = J\) then the organiser will expect to make a loss. Note: \(\e > 2\).
  2. Instead of the numbers being equally popular, a fraction \(\gamma\) of the numbers are popular and the rest are unpopular. For each participant, the probability of picking any given popular number is \(\dfrac{a}{N}\) and the probability of picking any given unpopular number is \(\dfrac{b}{N}\,\). Find a relationship between \(a\), \(b\) and \(\gamma\). Show that, using the approximation \((*)\), the organiser's expected profit can be expressed in the form \[ A\e^{-a} + B\e^{-b} +C \,, \] where \(A\), \(B\) and \(C\) can be written in terms of \(J\), \(c\), \(N\) and \(\gamma\). In the case \(\gamma = \frac18\) and \(a=9b\), find \(a\) and \(b\). Show that, if \(2Nc = J\), then the organiser will expect to make a profit. Note: \(\e < 3\).

Solution:

  1. The probability no-one picks the winning number is \(\left ( 1 - \frac{1}{N}\right)^N \approx \frac1e\). \begin{align*} && \mathbb{E}(\text{profit}) &= Nc - (1-e^{-1})J \\ &&& < Nc -(1- \tfrac12 )J \\ &&& < Nc - \frac12 J \\ &&&= \frac{2Nc-J}{2} \end{align*} Therefore if \(J = 2Nc\) the expected profit is negative.
  2. \(\,\) \begin{align*} && 1 &= \sum_{\text{all numbers}} \mathbb{P}(\text{pick }i) \\ &&&= \sum_{\text{popular numbers}} \mathbb{P}(\text{pick }i)+\sum_{\text{unpopular numbers}} \mathbb{P}(\text{pick }i) \\ &&&=\gamma N \frac{a}{N} + (1-\gamma)N \frac{b}{N} \\ &&&= \gamma a + (1-\gamma)b \end{align*} \begin{align*} && \mathbb{P}(\text{no-one picks winning number}) &= \mathbb{P}(\text{no-one picks winning number} | \text{winning number is popular})\mathbb{P})(\text{winning number is popular}) + \\ &&&\quad + \mathbb{P}(\text{no-one picks} | \text{unpopular})\mathbb{P}(\text{unpopular}) \\ &&&= \left (1 - \frac{a}{N} \right)^N \gamma + \left (1 - \frac{b}{N} \right)^N (1-\gamma) \\ &&&\approx \gamma e^{-a} + (1-\gamma)e^{-b} \\ \\ && \mathbb{E}(\text{profit}) &= Nc - (1-\gamma e^{-a} - (1-\gamma)e^{-b})J \\ &&&= Nc-J+J\gamma e^{-a} +J(1-\gamma)e^{-b} \end{align*} If \(\gamma = \frac18\) and \(a=9b\), then \(1=\frac18 a + \frac78b = 2b \Rightarrow b = \frac12, a = \frac92\) and \begin{align*} && \mathbb{E}(\text{profit}) &= Nc-J +J\tfrac18e^{-9/2}+J\tfrac78e^{-1/2} \\ &&&= Nc-J+\tfrac18Je^{-1/2}(e^{-4}+7) \end{align*} If we can show \(e^{-1/2}\frac{e^{-4}+7}{8} > \frac12\) we'd be done, so \begin{align*} && e^{-1/2}\frac{e^{-4}+7}{8} &> \frac12 \\ \Leftrightarrow && e^{-4}+7 &>4e^{1/2} \\ \Leftrightarrow && 49+14e^{-4}+e^{-8} &>16e \\ \end{align*} But clearly the LHS \(>49\) and the RHS \(<48\) so we're done

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2001 Paper 2 Q12
D: 1600.0 B: 1484.0
probability Poisson approximation binomial distribution expected value lottery optimisation most probable value approximating binomial to Poisson

The national lottery of Ruritania is based on the positive integers from \(1\) to \(N\), where \(N\) is very large and fixed. Tickets cost \(\pounds1\) each. For each ticket purchased, the punter (i.e. the purchaser) chooses a number from \(1\) to \(N\). The winning number is chosen at random, and the jackpot is shared equally amongst those punters who chose the winning number. A syndicate decides to buy \(N\) tickets, choosing every number once to be sure of winning a share of the jackpot. The total number of tickets purchased in this draw is \(3.8N\) and the jackpot is \(\pounds W\). Assuming that the non-syndicate punters choose their numbers independently and at random, find the most probable number of winning tickets and show that the expected net loss of the syndicate is approximately \[ N\; - \; %\textstyle{ \frac{5 \big(1- e^{-2.8}\big)}{14} \;W\;. \]

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2000 Paper 3 Q12
D: 1700.0 B: 1553.7
probability Poisson distribution binomial distribution conditional probability lottery approximation expected value large N limit

In a lottery, any one of \(N\) numbers, where \(N\) is large, is chosen at random and independently for each player by machine. Each week there are \(2N\) players and one winning number is drawn. Write down an exact expression for the probability that there are three or fewer winners in a week, given that you hold a winning ticket that week. Using the fact that $$ {\biggl( 1 - {a \over n} \biggr) ^n \approx \e^{-a}}$$ for \(n\) much larger than \(a\), or otherwise, show that this probability is approximately \({2 \over 3}\) . Discuss briefly whether this probability would increase or decrease if the numbers were chosen by the players. Show that the expected number of winners in a week, given that you hold a winning ticket that week, is \( 3-N^{-1}\).

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