Introduction

The following article appeared in a newspaper in Stuttgart on 29th June 1995:

Stuttgart  (dpa/lsw)  The  State  Toto-Lotto  GmbH in Stuttgart has  re- ported  a  lottery  sensation:   For  the first  time  in  the  40-year  history  of the  German  ‘Zahlenlotto’,  two  identical series  were  drawn.   On  the  21st June  of this  year the  series  15-25-27- 30-42-48 was  drawn in  the  lottery. Exactly the same series was drawn at on 20th December 1986.

The German ‘Zahlenlotto’ has 49 balls, numbered 1 to 49, and 6 balls are drawn without being replaced. Thus there are

(6(49)) = 13, 983, 816 ≈ 14, 000, 000

possible outcomes. There is one draw each Saturday, so the combination above was drawn twice after the lottery had only had about 2,000 draws! The aim of this project is to study whether this event is really as unlikely as it sounds at first.

Assessment

This project contributes 25% of the overall assessment for MATH240 Project Skills.  You must submit a report, expressed in your own words. Identical content (text, tables, figures et  cetera) in reports from different individuals may be rewarded with only a share of the available marks. Assessment will be based solely on the written report.

Your report should aim to present an investigation of the ‘paradox’in the title. You should write your report using LATEX.

A clear and concise report of 4 pages is fine, up to a limit of 6 pages, including figures.  You should be selective in the examples and illustrations that you use.  A good mark will be gained by a well-structured and well-presented report that includes a balanced account of experimentation and mathematical investigation. Don’t swamp the report with graphs of sequences, and don’t rely on these to tell the whole story.  Present clearly the points that you have learned from your investigation and that you wish to communicate to the reader. Try to attract and retain their interest.

Deadline and submission arrangements

The deadline for submission is 11:59pm on Friday of week 9, 9th December. You should submit your project in pdf format on Moodle. Office hours will be 4-5pm on Mondays in Weeks 6-9 in Fylde B57 office.  If you are unavailable at these times then please e-mail Gabor to arrange an alternative time. His e-mail address is: [email protected].

Detailed Project Description

A problem closely related to lotto-paradox  mentioned above is the birthday paradox.  This paradox is about the probability that in a group of k randomly chosen people some pair of them will have the same birthday. Since there are only 366 possible birthdays (including the 29th February), we immediately see that the probability is 1 if we have k = 367. Surprisingly, this probability is already very large for relatively small k . Indeed, the probability is 0.5 for k = 23 and 0.999 for k = 70.

The lotto-paradox and birthday paradox have similar setting. In both cases, we have a random experiment with n possible outcomes. For the birthday paradox, we have n = 366 and for lotto paradox, we have n = (6(49)). We assume at this point that each outcome has the same probability to occur1 . We now number the possible outcomes from 1 to n and repeat this experiment independently and infinitely often (at least theoretically). This could give for instance the sequence

5, 3, 4, 6, 9, 1, 5, 9, . . .                                                   (1)

In view of the lotto-paradox  and birthday paradox, we are interested in the first repetition in the obtained sequence. For this we introduce the random variable

X n)  := the number of repetitions needed till one obtains a repeated outcome.

We have for instance X n)  = 7 for the sequence in (1).  The aim of this project is study the properties of the random variable Xn)   and especially the probability P [X n)  ≤ k].

Questions you can address in your report are the following:

❼ Determine an exact formula for P [X n)  ≤ k].

Hint:  The Π notation for products uj(k)=1 (1 + aj ) = (1 + a1 )(1 + a2 ) ··· (1 + ak ) can be helpful to simplify the  expression for P [X n)  ≤ k] .

❼ Draw a graph of P [X n)  ≤ k] as a function of k for n fixed for several values of n,

and to compare the graphs for different n.

❼ Use the  computer  (R  or  Excel  or  . . . )   to  determine  numerically  a  k  such that

P [X n)  ≤ k] = 0.95 for several values of n, including n = 366 (birthday paradox) and for n = (6(49)) (lotto paradox).

❼ Use the  approximation  e −x    ≈  1 − x  for  x  small to  derive  an  approximation  of

P [X n)  ≤ k] for k small compared to n and then to use the computer to compare

this approximation with the exact value.

❼ Determine the behaviour of E [X n)] and Var (X n)) (numerically and/or theoretical

exact).

❼ Determine the joint distribution X n)  and Xn), where Xn)  is the number of repeti- tions needed till one obtains a second time a repeated outcome.  Are X n)  and Xn) independent or are X n)  and Xn) − X n)  independent?

You do not have to address all of these points in your report. You are of course also  allowed to address other questions related to Xn)  and P [X n)  ≤ k], which are not listed above.