Euler's Analysis of the Genoese Lottery - Classroom Activities

Here are a few classroom activities which might be incorporated into a lesson plan on lotteries and probability theory. Depending on the approach you take, a presentation of Euler's methods for calculating lottery properties could make for a good topic for any of the grades 9–14.

• Historical Background. Discussion of any of the following:
   
  • The life and work of Euler (see [3, 2, 4])
     
  • The court of Frederick the Great
     
  • The history of probability theory up to Euler's time, and especially the special place of Pascal's triangle in early probability (see [9, 1])
     
  • The history of lotteries (see [7])
     
• Real-world example. A brief presentation on a lottery game that is actually played in your community (e.g. NYS Lotto, Powerball). What are the parameters \(n\) and \(t\)? Do players have the same choice of bets as in the Genoese Lottery? Are there any modern elaborations (e.g bonus numbers)?
   
• Small-scale model. Design of a small-scale lottery; for example \(n=10\) or even \(8\), and \(t=3\). The numbers here are manageable. Do the students have any intuition about how likely it is to match 1, 2 or 3 numbers? Since modern lotteries often only permit bets on \(k=t\) numbers, this is also a way to introduce the notion of a lottery where the players may choose to bet on only one or two numbers. Is this sort of modest betting what people want, or is the natural tendency always to go for the jackpot?
   
• Calculations. Calculate the probabilities for a the small-scale lottery. We only understand what Euler's formulas mean when we use them. There are 3 steps:
  • Calculate Pascal's triangle for your choice of \(n\) and \(t\); that is Pascal's triangle with \(t\) rows. For example:
    \[ \begin{array}{c} 1 \quad 1 \\ 1 \quad 2 \quad 1 \\ 1 \quad 3 \quad 3 \quad 1 \\ \end{array} \]
  • Calculate the array of \(s_{k,i}\)s using Euler's recursive rules. As in Pascal's triangle, the first and last elements of each row are special, but each internal element is simply formed by looking at the two elements above it: the factors of the numerator consist of the union of the numerators above, and the denominator has one additional factor.
    \[ \begin{array}{c} {\frac{7}{10}} \quad {\frac{3}{10}} \\ {\frac{7 \cdot 6}{10 \cdot 9}} \quad {\frac{7 \cdot 3}{10 \cdot 9}} \quad {\frac{3 \cdot 2}{10 \cdot 9}} \\ {\frac{7 \cdot 6 \cdot 5}{10 \cdot 9 \cdot 8}} \quad {\frac{7 \cdot 6 \cdot 3}{10 \cdot 9 \cdot 8}} \quad {\frac{7 \cdot 3 \cdot 2}{10 \cdot 9 \cdot 8}} \quad {\frac{3 \cdot 2 \cdot 1}{10 \cdot 9 \cdot 8}} \\ \end{array} \]
  • Calculate the array of probabilities \(p_{k,i}\) by multiplying these two triangles, term by term.
   

• Real-life Analysis. Calculate the probabilities for the lottery that is actually played in your region. Compare the calculated probabilities to actual recent payoffs; this will give the students an idea of the magnitude of the bank's margin. An applet will appear on this site soon to help with your calculations.  Come back later to use it.


• Weighting and Fairness If available, find out how prize money is distributed in your regional lottery, especially the size of the bank's margin. What are the proceeds used for? Does the weighting scheme look anything like the schemes considered by Euler?

Editor’s note: This article was published in April 2004.