2001 Paper 2 Q12

Year: 2001
Paper: 2
Question Number: 12

Course: UFM Statistics
Section: Approximating the Binomial to the Poisson distribution

Difficulty: 1600.0 Banger: 1484.0
probability Poisson approximation binomial distribution expected value lottery optimisation most probable value approximating binomial to Poisson

Problem

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 \[ N\; - \; %\textstyle{ \frac{5 \big(1- e^{-2.8}\big)}{14} \;W\;. \]

No solution available for this problem.

Rating Information

Difficulty Rating: 1600.0

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Banger Rating: 1484.0

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Problem source
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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