- Email: [email protected]
The Lottery of Casanova
6.1 THE LOTTERY OF CASANOVA Hans-Wolfgang H e m and Andreas Buchter University of Dortmund, Germany Abstract-This paper presents the results and experiences of a project we did at Dortmund University with students in our teacher education department. The open-ended task to analyse the “Lottery of Casanova ”provided a productive learning experience and leads naturally to various modelling approaches. These diferent approaches emerge through dzflerent perspectives of the situation. The context opens up a large potential for the design of modem mathematics teaching, oriented at current didactical concepts starting fi-om phenomena and leading to the formation of mathematical theories.
1. AUTHENTIC STOCHASTIC MODELLING The Lottery of Genoa or “Lottery of Casanova”, as we call it, which will be described, has been used within our project, “authentic stochastic modelling”. Our goal was the development of learning environments for stochastic modelling for prospective primary and lower-secondary mathematics teachers. Our theoretical background was Wittmann’s constructivistic approach to mathematics education as a “design science” (Wittmann, 1995). The project was embedded in a lecture series “Introduction to Stochastics” and “Didactics of Stochastics”, which is usually taken by students in two consecutive semesters. Freudenthal (1983) characterizes stochastics as a classic example of realistic mathematics having many associations. Although some “simple mathematics” can be sufficient for stochastics modelling, it is often not clear how this simple mathematics has to be used. Typical situations are: 0 Different solution strategies seem to be equally plausible, but lead to different results. Different solution strategies lead to the same result, but it is not obvious, why? Working with such “paradoxa” is a suitable method to develop “Grundvorstellungen” (basic ideas) of stochastic concepts and modelling competence. We will illustrate this by using the “Lottery of Casanova” which was part of this project. Known worldwide and recently introduced in Germany, the Keno lottery has more or less the same structure as the historic Genoa lottery. The mathematical treatment with both lotteries is a paradigmatic example of authentic stochastic modelling. The open-ended task, to analyse the lotteries, provides a productive learning experience and leads naturally to various modelling approaches. These different approaches emerge through different perspectives of the situation. The context opens up a large potential for the design of modern mathematics teaching, oriented at current didactical concepts, starting from phenomena and
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leading to the formation of mathematical theories. The analysis of the structurally related Keno lottery proves to be a suitable tool to check the learning achievement of students in assessment situations. 2. THE LOTTERY OF GENOA The origin of the Lottery of Genoa lies in the election of senators, which was introduced in 1575 following a coup d’ttat (Kratz & Merlin 1995, p6.5). Five citizens from a list of 90 were given the senator status to complement the existing council. Resulting from various bids on which citizens would rise to senator status, a lottery developed in Italy until the year 1643. This Lottery of Genoa was introduced to many European countries. In 1758, the Venetian adventurer Casanova (1725 1798) introduced a lottery similar to the model of the Lottery of Genoa in France. In this lottery five winning numbers were drawn out of ninety. The participants could put one number, two numbers (Ambe) or three (Terne) on their coupon. One drawn winning number was reimbursed with a multiple of fifteen of the stakes, two with a multiple of 270, and three winning numbers with a multiple of 5,200 of the stakes (Childs 1961). Table 1 gives an overview of the different options.
Options Numbers chosen Drawing pay-out
Single number Ambe Terne 1 2 3 5 out of 90 balls are drawn 270 times 5,200 times 15 times
I
~~
1
~
~
Table 1. Options of the Lottery of Genoa. 3. DIFFERENT SOLUTION STRATEGIES GIVE DIFFERENT RESULTS The lecture started with the fundamentals of descriptive statistics. Afterwards, beginning with the theme ‘probability’ a short introduction to the Lottery of Genoa was given, students were then asked to analyze this lottery. In small laboratory course groups students had to choose questions to work on. One question could be chosen by more than one group. All student groups started working on their chosen questions by trying to determine winning probabilities for the three lottery options. The most popular approach, presented here for “Terne”,was the following:
For the denominator the students started - close to the situation - from the number of possibilities to draw five out of 90 numbers, oriented on the drawing process. For the numerator they calculated with the number of chosen and the number of drawn numbers. The argumentation usually went as follows: “Three numbers have to appear withinJive numbers, therefore two are left over which have to be among the other 85 numbers”. Reflecting on this approach, students discussed
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how this approach is different to a fictional game option where five numbers are chosen of which three are then among the drawn numbers. For this fictive option the same formula holds. Obviously, it is more difficult to choose three numbers, which are then all drawn, compared to choose five numbers, out of which three are then drawn. After that, the students tried to save the above approach by the procedure:
I:(
P(Teme) = -z 0.00002%
(3
Further reflection on this attempted rescue lead to more objections. If this approach was correct, the chance to choose three winning numbers would be the same as the chance to choose two winning numbers:
On the basis of these considerations the students realized that this approach to model the Lottery of Genoa led to a dead end. 4. DIFFERENT STRATEGIES LEAD TO THE SAME RESULT To come out of the dead end the students started to discuss the (to them) better known German lottery “6 out of 49”, which led to hrther solution strategies in the small working groups. Again, we look at the “Terne“. 131 187)
Approach A: P(Terne) = (3J’( (90)
=1s0.00851% 11,748
To calculate this winning probability the drawing of the numbers is - close to the situation - modelled with the Laplace approach as a stochastic experiment. Five numbers are drawn out of 90. The denominator is the number of possible results. Taking into consideration that three of the numbers drawn have to coincide with the three chosen numbers and that the other two drawn numbers have to come from the 87 not chosen numbers, gives the term for the numerator. Another interpretation of the same model was: Choosing colours three balls from an urn holding 90 balls as personal winning balls. What now is the probability to get these three balls by drawing five balls without replacement? 15)
Approach B: P(Terne) = -(90) - 11,748
The second Laplace approach for modelling the Lottery of Genoa looks at the possible and the favourable chosen numbers. In spite ofthe chronology of the lottery the assumption is made that the drawn numbers are fixed and the process of doing
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the lottery is taken as the stochastic experiment. Three out of 90 numbers are chosen; therefore the denominator counts the number of possible choices. For the numerator the three chosen numbers need to be among the five drawn numbers. In the alternative way of thinking this means: The later drawing colours five balls from 90 as winning balls. How big is the probability by choosing three different numbers (model of drawing without replacement) to get three out of the five coloured balls? 5 4 3 1 Approach C: P(Terne) = -.-.== 0.00851%. 90 89 88 11,748 A third, also discussed, modelling approach works directly with the alternative interpretation. Again - against the chronology of the lottery - the model is based on an urn with five “winning balls”. The process of doing the lottery, here the drawing of three balls without replacement, is calculated: For putting the first number on one‘s coupon the chance to gain a winning ball results from the ratio of five winning balls to 90 balls altogether. In the same way the chances for the second and third number are found. The resulting probability for winning is the product of the three fractions according to the path rule. A look at the three reduced fractions for the three approaches (A, B, C) and the winning percentages shows that the numerical check of all three approaches results in the same chance. The correspondence of the solutions can be seen as the mutual validation of the three solution strategies. Students, who prefer one of the approaches and have no insight in the other approaches, can be convinced that the other approaches also work. They can be motivated to try to understand the other approaches, too. The question “why” becomes interesting and leads to a structural validation of the different solution approaches.
5. WHAT DOES THE STATE EARN? Following the introduction of the random variable concept the Lottery of Genoa was discussed again. Emphasis was given to the question of most interest for the organizer of the lottery: “How big is the expected profit?” For an intuitively apparent idea, which will later become the expected value, the stakes are chosen to be I € per bet. The winning probability for the single-number-bet is For a “normal” course there is exactly one win of 15 € per 18 players in the long run. That means, 18 times the state receives I €, one time the state will have to pay out 15 €. If 18 players have placed a single-number-bet EP(sing1e number) = 18.1 € - 1.15 € = 3 € will be the expected profit for the state. The abbreviation “EP” stands for “expected profit”. Analogous considerations for the other two possibilities to bet result in the profit expectations EP(Ambe) = 801.1 € - 2.270 € = 261 €, EP(Terne) = 1 1,748.1 € - 1.5,200 € = 6,548 €. Now it would be incorrect to deduct that the bid “Terne” is the most profitable for the state, because the analysis is based on different numbers of participating gamblers. The necessary remedy is obvious: The comparison must be based on the relative profit per gambler
Henn and Buchter ep(sing1e number) =
363
EP(sing1e number) = 0.17€ , 18
ep(Ambe) = EP(hbe) FZ 0.33 e , 80 1 EP(Teme) ep(Terne) = = 0.56€. 11,748 From the point of view of the state the option “Teme” is indeed the most profitable, if everything runs smoothly. This is also the option which will - because of the large possible profit - attract most gamblers. But the game could not run smoothly and it might happen that about 10% of all gamblers have chosen the winning numbers. As the profit margin is a fixed number (5,200 times the stakes), the bank might collapse. To analyse the question “How big is the expected profit for the state?” sensibly further considerations are necessary. If we are interested in the profit which the state can make as additional income, further information have to be available or further assumptions have to be made. These normative actions consist in the decision how many gamblers can participate in the lottery, how big the stakes are, how many of them choose which option of the lottery and which spread of winners is to be expected for the three different options. Suitable methods for these issues can be developed easily. For this, we analyse in detail the relative profit per gambler for the single number bid from the viewpoint of the state’s profit and loss probability: If the state accepts a single number bid, then it has to pay 14 € with the probability and with the probability l7Il8 it wins 1 €. The single number bid can be described as a random variable ZI, measured in €, that can take the values -14 and 1 with a probability of P(-14) = and P(l) = 17/,8. The expected profit of the state per gambler, measured in €, can then be described by 17.1- 1.14 EP(sing1e number) E(Z1) := ep(sing1e number) = 18 18 1 17 1 = --.1+--.(-14) = P(Z, = 1).1+ P(2, = -14). (-14) = - = 0.17. 18 18 6 What we have found is the concept of the expected value E(Z) of a random variable Z. We compare this approach with the calculation of the arithmetical mean of a sample using relative frequencies: If 1,000 gamblers bid on one number and 50 of them have won the balance of the state, again in €, will be 950.1-50.14 950 1+-. 50 - -. (-14) = 0.25. Average profit per gambler = 1,000 1,000 1,000 Substituting the relative frequencies in the formula for the arithmetic mean of a sample by the probability of a random variable, leads to the expected value of the random variable. Analogously, the expected values for the two other options are: 26 1 6 548 Ambe: E(2,) = -= 0.33 ; Teme: E(Z,) = -= 0.56 80 1 11,748 The positive expected value of the Lottery of Genoa obviously does not guarantee a profit for the state. If, by chance, among 18 gamblers there are two winners, the state looses 16.1 € - 2.14 € = -12 €. A sensible analysis calls for a
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prediction of the spread to be expected. Our naive approach for the expected value has lead to a formula which is analogue to the arithmetic mean. Accordingly, variance V(Z) and standard deviation oz are introduced. These definitions, again, are analogue to the definitions of empirical variance and standard deviation of samples, where probabilities take the place of relative frequencies. V(Z,) := (1 - E(Z,))‘ .P(Z, = 1) + (-14 - E(Z,))’ .P(Z, = -14) 11.81 and ciz, :=
-
Jqq)3.44.
-
The corresponding values for the other two options are Ambe: oz, -13.47; Terne: oz, -47.97. Compared with the expected value the value of the standard deviation is rather large and points at the risk for the state. In order to come to more precise statements about the “Lottery of Casanova”, the topic was re-introduced for a third time at the end of the lecture course in relation to the binomial and normal distributions. Some of the students proposed the following, again shown using the option Terne: One bet is looked upon as a Bernoulli experiment with the possible results 0 = “the state wins” and 1 = “the gambler wins”. The corresponding random variable therefore possesses the values P(X=O) = 1 1 1,747 and P(X=l) = -=: p . We consider a lottery where n gamblers want to 1 1,748 11,748 play Terne. Together with the sensible model assumption that all gamblers bet independently from each other, the situation can then be described as a random variable Z following a binomial distribution with parameters p and n. The state receives 1 Euro from each gambler and pays 5,200 Euro to each winner. If m is the number of winners a profit for the state is only possible if n - 5,200.m 2 0. For a n given n there should not be more than m* := trunc(-) winners. The risk for the 5,200 state is that there are more than m* winners. For that the probability is given by
Trying to evaluate this formula for concrete values of n using MAPLEor any other CAS, the limits of the computer are reached fast (by only calculating the sum from 0 to m*, too.). So, the normal distribution has to be used as an approximation for the binomial distribution. Using the expected value p= n.p and the standard of the binomial distribution, the following formula has deviation ci = to be evaluated
= /,
P(Z>m*)= J -e 2 m* &.o.
This is a much easier task for MAPLE.For example, the’risk probability for n = 100,000 is approximately 0.00041, for n = 1,000,000 this probability is 0 (calculated with a precision of 20 digits).
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Henn and Buchter 6. THE KENO LOTTERY
The Keno Lottery, introduced in 2004 in some German federal states is a good real life example for mathematics teaching. The lottery has several aspects which are of interest for teaching. It is the first virtual lottery in Germany. Winning numbers are drawn by a computer, which has been developed especially for this purpose by the Fraunhofer Institut f i r Rechnerarchitektur und Softwaretechnik - a fact that motivates to reflect on the generation of “random numbers” by computers. The Keno Lottery is played mostly via the Internet and aims at adolescents and young adults. Keno is probably the world’s oldest lottery. More than 2,000 years ago it was already played in China as White Doves Game. Since quite some time Keno is very popular in Anglo-Saxon countries. From a stochastic point of view, Keno’s structure is similar to the Lottery of Genoa. 20 out of 70 numbers are drawn. Gamblers have to decide to play one of the options Kenotype 2 to Kenotype 10 and then choose two to ten numbers, respectively, Figure 1 shows a lottery coupon of the German Keno Lottery which allows the gambler 5 different bets. The gambler has to mark with a cross the chosen Kenotype (“Anzahl getippter Zahlen”) for every bet and then to mark the desired numbers with a cross, for example 5 numbers when the Kenotyp 5 is chosen. As stakes gamblers can choose (with a cross in “Einsatz”) 1 € , 2 €, 5 € or 10 €. “plus 5” is an additional game, a cross in “Anzahl der Ziehungen” indicates the duration (in weeks) of validity of the coupon. I 1 1 1 1 1 1 1 if I I I I I 1 1 1 1 1 1 1 I I I I l l l I I I I I I l f
I
I*_--
..,,..*
1*%
.I*“
... .
1
I
Figure 1 . The German Keno Lottery. In each case the number of numbers chosen and numbers drawn are different. Contrary to the Lottery of Genoa the gambler does not only win if all chosen numbers are drawn. In the options with eight, nine or ten numbers to choose (Kenotype 8, 9, and lo), the gambler also wins if none of his or her numbers are selected. Therefore, Keno is the first lottery where unlucky fellows win. Altogether there are 36 prize categories. And, obviously, there is a lot to calculate. On the reverse side of the Keno Coupon (Figure 2) all winning classes are described. The first column shows the Kenotyp and the second the number of correctly chosen numbers. The columns 3 to 6 show the profit for the stakes 1 €, 2 €, 5 €, and 10 €.
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The Lottery of Casanova
As for the Lottery of Genoa the Keno Lottery has fixed winning ratios, between one and 100,000 times the stakes. Theoretically the lottery can turn out a loss for the provider. To avoid this, the two highest winning categories have an additional provision. It states that in Kenotype 10 for 10 correctly predicted numbers a win of 100,000 times the stakes is only given to five winners. If more than five gamblers win in this category the maximum sum is divided among all those winners. Similar rules apply to Kenotype 9 for 9 correctly predicted numbers, with a maximum number of ten winners. The actuality of this lottery motivates students to deal with it in detail. For an analysis of the winning situation described in Figure 2 we define the event mJn:“a hit rate of m correctly predicted numbers in Kenotype n” I
Figure 2. The prize categories. Now the winning possibilities from the perspective of the numbers drawn can be deduced easily. For example, to have 8 hits for Kenotype 10, 8 of the 20 winning numbers and 2 of the non-winning numbers have to be marked. This leads to the probability
For the same Kenotype 10, the probability to have chosen none of the winning numbers is only
150)
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Henn and Buchter Generally, the probability to have m hits in Kenotype n is: P(m I n) =
(z)
(n lorn) (70) *
The largest single winning probability is reached for 2 hits in Kenotype 3 with P(213) = 0.174. The probability to win anything at all in Kenotype n is given by P(Win 1 Type n) =
2
P(m I n) .
m=O
rn leads to profit
This winning probability takes its maximum for Kenotype 4 with P(Win I Type 4) = 0.321 and its minimum for Kenotype 2 with P(Win I Type 2) = 0.079. As for the “Lottery of Casanova” an interesting task for Keno is to analyze the state’s risk for each of the game options. 7. CONCLUSION This paper on making use of the “Lottery of Casanova” in university teacher education shows some aspects of problem contexts which have a positive effect for productive learning environments. To design a teaching-learning environment according to current conceptions in mathematics education contexts including these aspects have to be developed. Searching for and developing such contexts is therefore essential for the further development of mathematics education. But a good context alone is not sufficient. It has to be used for good teaching, which is not always easy in a concrete teaching-learning situation. It is important that enough time is planned for individual learning and the reflection on this processes, as well as for different solution strategies and their discussion.
REFERENCES Buchter, A. and Henn, H.-W. (2004) Stochastische Modellbildung aus unterschiedlichen Perspektiven - Von der Genueser Lotterie uber Urnenaufgaben zur Keno Lotterie. Stochastik in der Schule, 24 (3), 28-4 1. Childs, J.R. (1961) Casanova: A Biography Based on New Documents. London: George Allen and Unwin Ltd. Freudenthal, H. (1 983) Didactical phenomenology of mathematical structures. Dordrecht: Reidel. Kratz, 0. and Merlin, H. (1995) Casanova. Liebhaber der Wissenschaften. Munchen: Callwey. Wittmann, E.Ch. (1995) Mathematics education as a “design science”. Educational Studies in Mathematics, 29,355-374.
1. AUTHENTIC STOCHASTIC MODELLING The Lottery of Genoa or “Lottery of Casanova”, as we call it, which will be described, has been used within our project, “authentic stochastic modelling”. Our goal was the development of learning environments for stochastic modelling for prospective primary and lower-secondary mathematics teachers. Our theoretical background was Wittmann’s constructivistic approach to mathematics education as a “design science” (Wittmann, 1995). The project was embedded in a lecture series “Introduction to Stochastics” and “Didactics of Stochastics”, which is usually taken by students in two consecutive semesters. Freudenthal (1983) characterizes stochastics as a classic example of realistic mathematics having many associations. Although some “simple mathematics” can be sufficient for stochastics modelling, it is often not clear how this simple mathematics has to be used. Typical situations are: 0 Different solution strategies seem to be equally plausible, but lead to different results. Different solution strategies lead to the same result, but it is not obvious, why? Working with such “paradoxa” is a suitable method to develop “Grundvorstellungen” (basic ideas) of stochastic concepts and modelling competence. We will illustrate this by using the “Lottery of Casanova” which was part of this project. Known worldwide and recently introduced in Germany, the Keno lottery has more or less the same structure as the historic Genoa lottery. The mathematical treatment with both lotteries is a paradigmatic example of authentic stochastic modelling. The open-ended task, to analyse the lotteries, provides a productive learning experience and leads naturally to various modelling approaches. These different approaches emerge through different perspectives of the situation. The context opens up a large potential for the design of modern mathematics teaching, oriented at current didactical concepts, starting from phenomena and
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The Lottery of Casanova
leading to the formation of mathematical theories. The analysis of the structurally related Keno lottery proves to be a suitable tool to check the learning achievement of students in assessment situations. 2. THE LOTTERY OF GENOA The origin of the Lottery of Genoa lies in the election of senators, which was introduced in 1575 following a coup d’ttat (Kratz & Merlin 1995, p6.5). Five citizens from a list of 90 were given the senator status to complement the existing council. Resulting from various bids on which citizens would rise to senator status, a lottery developed in Italy until the year 1643. This Lottery of Genoa was introduced to many European countries. In 1758, the Venetian adventurer Casanova (1725 1798) introduced a lottery similar to the model of the Lottery of Genoa in France. In this lottery five winning numbers were drawn out of ninety. The participants could put one number, two numbers (Ambe) or three (Terne) on their coupon. One drawn winning number was reimbursed with a multiple of fifteen of the stakes, two with a multiple of 270, and three winning numbers with a multiple of 5,200 of the stakes (Childs 1961). Table 1 gives an overview of the different options.
Options Numbers chosen Drawing pay-out
Single number Ambe Terne 1 2 3 5 out of 90 balls are drawn 270 times 5,200 times 15 times
I
~~
1
~
~
Table 1. Options of the Lottery of Genoa. 3. DIFFERENT SOLUTION STRATEGIES GIVE DIFFERENT RESULTS The lecture started with the fundamentals of descriptive statistics. Afterwards, beginning with the theme ‘probability’ a short introduction to the Lottery of Genoa was given, students were then asked to analyze this lottery. In small laboratory course groups students had to choose questions to work on. One question could be chosen by more than one group. All student groups started working on their chosen questions by trying to determine winning probabilities for the three lottery options. The most popular approach, presented here for “Terne”,was the following:
For the denominator the students started - close to the situation - from the number of possibilities to draw five out of 90 numbers, oriented on the drawing process. For the numerator they calculated with the number of chosen and the number of drawn numbers. The argumentation usually went as follows: “Three numbers have to appear withinJive numbers, therefore two are left over which have to be among the other 85 numbers”. Reflecting on this approach, students discussed
Hem and Buchter
361
how this approach is different to a fictional game option where five numbers are chosen of which three are then among the drawn numbers. For this fictive option the same formula holds. Obviously, it is more difficult to choose three numbers, which are then all drawn, compared to choose five numbers, out of which three are then drawn. After that, the students tried to save the above approach by the procedure:
I:(
P(Teme) = -z 0.00002%
(3
Further reflection on this attempted rescue lead to more objections. If this approach was correct, the chance to choose three winning numbers would be the same as the chance to choose two winning numbers:
On the basis of these considerations the students realized that this approach to model the Lottery of Genoa led to a dead end. 4. DIFFERENT STRATEGIES LEAD TO THE SAME RESULT To come out of the dead end the students started to discuss the (to them) better known German lottery “6 out of 49”, which led to hrther solution strategies in the small working groups. Again, we look at the “Terne“. 131 187)
Approach A: P(Terne) = (3J’( (90)
=1s0.00851% 11,748
To calculate this winning probability the drawing of the numbers is - close to the situation - modelled with the Laplace approach as a stochastic experiment. Five numbers are drawn out of 90. The denominator is the number of possible results. Taking into consideration that three of the numbers drawn have to coincide with the three chosen numbers and that the other two drawn numbers have to come from the 87 not chosen numbers, gives the term for the numerator. Another interpretation of the same model was: Choosing colours three balls from an urn holding 90 balls as personal winning balls. What now is the probability to get these three balls by drawing five balls without replacement? 15)
Approach B: P(Terne) = -(90) - 11,748
The second Laplace approach for modelling the Lottery of Genoa looks at the possible and the favourable chosen numbers. In spite ofthe chronology of the lottery the assumption is made that the drawn numbers are fixed and the process of doing
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The Lottery of Casanova
the lottery is taken as the stochastic experiment. Three out of 90 numbers are chosen; therefore the denominator counts the number of possible choices. For the numerator the three chosen numbers need to be among the five drawn numbers. In the alternative way of thinking this means: The later drawing colours five balls from 90 as winning balls. How big is the probability by choosing three different numbers (model of drawing without replacement) to get three out of the five coloured balls? 5 4 3 1 Approach C: P(Terne) = -.-.== 0.00851%. 90 89 88 11,748 A third, also discussed, modelling approach works directly with the alternative interpretation. Again - against the chronology of the lottery - the model is based on an urn with five “winning balls”. The process of doing the lottery, here the drawing of three balls without replacement, is calculated: For putting the first number on one‘s coupon the chance to gain a winning ball results from the ratio of five winning balls to 90 balls altogether. In the same way the chances for the second and third number are found. The resulting probability for winning is the product of the three fractions according to the path rule. A look at the three reduced fractions for the three approaches (A, B, C) and the winning percentages shows that the numerical check of all three approaches results in the same chance. The correspondence of the solutions can be seen as the mutual validation of the three solution strategies. Students, who prefer one of the approaches and have no insight in the other approaches, can be convinced that the other approaches also work. They can be motivated to try to understand the other approaches, too. The question “why” becomes interesting and leads to a structural validation of the different solution approaches.
5. WHAT DOES THE STATE EARN? Following the introduction of the random variable concept the Lottery of Genoa was discussed again. Emphasis was given to the question of most interest for the organizer of the lottery: “How big is the expected profit?” For an intuitively apparent idea, which will later become the expected value, the stakes are chosen to be I € per bet. The winning probability for the single-number-bet is For a “normal” course there is exactly one win of 15 € per 18 players in the long run. That means, 18 times the state receives I €, one time the state will have to pay out 15 €. If 18 players have placed a single-number-bet EP(sing1e number) = 18.1 € - 1.15 € = 3 € will be the expected profit for the state. The abbreviation “EP” stands for “expected profit”. Analogous considerations for the other two possibilities to bet result in the profit expectations EP(Ambe) = 801.1 € - 2.270 € = 261 €, EP(Terne) = 1 1,748.1 € - 1.5,200 € = 6,548 €. Now it would be incorrect to deduct that the bid “Terne” is the most profitable for the state, because the analysis is based on different numbers of participating gamblers. The necessary remedy is obvious: The comparison must be based on the relative profit per gambler
Henn and Buchter ep(sing1e number) =
363
EP(sing1e number) = 0.17€ , 18
ep(Ambe) = EP(hbe) FZ 0.33 e , 80 1 EP(Teme) ep(Terne) = = 0.56€. 11,748 From the point of view of the state the option “Teme” is indeed the most profitable, if everything runs smoothly. This is also the option which will - because of the large possible profit - attract most gamblers. But the game could not run smoothly and it might happen that about 10% of all gamblers have chosen the winning numbers. As the profit margin is a fixed number (5,200 times the stakes), the bank might collapse. To analyse the question “How big is the expected profit for the state?” sensibly further considerations are necessary. If we are interested in the profit which the state can make as additional income, further information have to be available or further assumptions have to be made. These normative actions consist in the decision how many gamblers can participate in the lottery, how big the stakes are, how many of them choose which option of the lottery and which spread of winners is to be expected for the three different options. Suitable methods for these issues can be developed easily. For this, we analyse in detail the relative profit per gambler for the single number bid from the viewpoint of the state’s profit and loss probability: If the state accepts a single number bid, then it has to pay 14 € with the probability and with the probability l7Il8 it wins 1 €. The single number bid can be described as a random variable ZI, measured in €, that can take the values -14 and 1 with a probability of P(-14) = and P(l) = 17/,8. The expected profit of the state per gambler, measured in €, can then be described by 17.1- 1.14 EP(sing1e number) E(Z1) := ep(sing1e number) = 18 18 1 17 1 = --.1+--.(-14) = P(Z, = 1).1+ P(2, = -14). (-14) = - = 0.17. 18 18 6 What we have found is the concept of the expected value E(Z) of a random variable Z. We compare this approach with the calculation of the arithmetical mean of a sample using relative frequencies: If 1,000 gamblers bid on one number and 50 of them have won the balance of the state, again in €, will be 950.1-50.14 950 1+-. 50 - -. (-14) = 0.25. Average profit per gambler = 1,000 1,000 1,000 Substituting the relative frequencies in the formula for the arithmetic mean of a sample by the probability of a random variable, leads to the expected value of the random variable. Analogously, the expected values for the two other options are: 26 1 6 548 Ambe: E(2,) = -= 0.33 ; Teme: E(Z,) = -= 0.56 80 1 11,748 The positive expected value of the Lottery of Genoa obviously does not guarantee a profit for the state. If, by chance, among 18 gamblers there are two winners, the state looses 16.1 € - 2.14 € = -12 €. A sensible analysis calls for a
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prediction of the spread to be expected. Our naive approach for the expected value has lead to a formula which is analogue to the arithmetic mean. Accordingly, variance V(Z) and standard deviation oz are introduced. These definitions, again, are analogue to the definitions of empirical variance and standard deviation of samples, where probabilities take the place of relative frequencies. V(Z,) := (1 - E(Z,))‘ .P(Z, = 1) + (-14 - E(Z,))’ .P(Z, = -14) 11.81 and ciz, :=
-
Jqq)3.44.
-
The corresponding values for the other two options are Ambe: oz, -13.47; Terne: oz, -47.97. Compared with the expected value the value of the standard deviation is rather large and points at the risk for the state. In order to come to more precise statements about the “Lottery of Casanova”, the topic was re-introduced for a third time at the end of the lecture course in relation to the binomial and normal distributions. Some of the students proposed the following, again shown using the option Terne: One bet is looked upon as a Bernoulli experiment with the possible results 0 = “the state wins” and 1 = “the gambler wins”. The corresponding random variable therefore possesses the values P(X=O) = 1 1 1,747 and P(X=l) = -=: p . We consider a lottery where n gamblers want to 1 1,748 11,748 play Terne. Together with the sensible model assumption that all gamblers bet independently from each other, the situation can then be described as a random variable Z following a binomial distribution with parameters p and n. The state receives 1 Euro from each gambler and pays 5,200 Euro to each winner. If m is the number of winners a profit for the state is only possible if n - 5,200.m 2 0. For a n given n there should not be more than m* := trunc(-) winners. The risk for the 5,200 state is that there are more than m* winners. For that the probability is given by
Trying to evaluate this formula for concrete values of n using MAPLEor any other CAS, the limits of the computer are reached fast (by only calculating the sum from 0 to m*, too.). So, the normal distribution has to be used as an approximation for the binomial distribution. Using the expected value p= n.p and the standard of the binomial distribution, the following formula has deviation ci = to be evaluated
= /,
P(Z>m*)= J -e 2 m* &.o.
This is a much easier task for MAPLE.For example, the’risk probability for n = 100,000 is approximately 0.00041, for n = 1,000,000 this probability is 0 (calculated with a precision of 20 digits).
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Henn and Buchter 6. THE KENO LOTTERY
The Keno Lottery, introduced in 2004 in some German federal states is a good real life example for mathematics teaching. The lottery has several aspects which are of interest for teaching. It is the first virtual lottery in Germany. Winning numbers are drawn by a computer, which has been developed especially for this purpose by the Fraunhofer Institut f i r Rechnerarchitektur und Softwaretechnik - a fact that motivates to reflect on the generation of “random numbers” by computers. The Keno Lottery is played mostly via the Internet and aims at adolescents and young adults. Keno is probably the world’s oldest lottery. More than 2,000 years ago it was already played in China as White Doves Game. Since quite some time Keno is very popular in Anglo-Saxon countries. From a stochastic point of view, Keno’s structure is similar to the Lottery of Genoa. 20 out of 70 numbers are drawn. Gamblers have to decide to play one of the options Kenotype 2 to Kenotype 10 and then choose two to ten numbers, respectively, Figure 1 shows a lottery coupon of the German Keno Lottery which allows the gambler 5 different bets. The gambler has to mark with a cross the chosen Kenotype (“Anzahl getippter Zahlen”) for every bet and then to mark the desired numbers with a cross, for example 5 numbers when the Kenotyp 5 is chosen. As stakes gamblers can choose (with a cross in “Einsatz”) 1 € , 2 €, 5 € or 10 €. “plus 5” is an additional game, a cross in “Anzahl der Ziehungen” indicates the duration (in weeks) of validity of the coupon. I 1 1 1 1 1 1 1 if I I I I I 1 1 1 1 1 1 1 I I I I l l l I I I I I I l f
I
I*_--
..,,..*
1*%
.I*“
... .
1
I
Figure 1 . The German Keno Lottery. In each case the number of numbers chosen and numbers drawn are different. Contrary to the Lottery of Genoa the gambler does not only win if all chosen numbers are drawn. In the options with eight, nine or ten numbers to choose (Kenotype 8, 9, and lo), the gambler also wins if none of his or her numbers are selected. Therefore, Keno is the first lottery where unlucky fellows win. Altogether there are 36 prize categories. And, obviously, there is a lot to calculate. On the reverse side of the Keno Coupon (Figure 2) all winning classes are described. The first column shows the Kenotyp and the second the number of correctly chosen numbers. The columns 3 to 6 show the profit for the stakes 1 €, 2 €, 5 €, and 10 €.
3 66
The Lottery of Casanova
As for the Lottery of Genoa the Keno Lottery has fixed winning ratios, between one and 100,000 times the stakes. Theoretically the lottery can turn out a loss for the provider. To avoid this, the two highest winning categories have an additional provision. It states that in Kenotype 10 for 10 correctly predicted numbers a win of 100,000 times the stakes is only given to five winners. If more than five gamblers win in this category the maximum sum is divided among all those winners. Similar rules apply to Kenotype 9 for 9 correctly predicted numbers, with a maximum number of ten winners. The actuality of this lottery motivates students to deal with it in detail. For an analysis of the winning situation described in Figure 2 we define the event mJn:“a hit rate of m correctly predicted numbers in Kenotype n” I
Figure 2. The prize categories. Now the winning possibilities from the perspective of the numbers drawn can be deduced easily. For example, to have 8 hits for Kenotype 10, 8 of the 20 winning numbers and 2 of the non-winning numbers have to be marked. This leads to the probability
For the same Kenotype 10, the probability to have chosen none of the winning numbers is only
150)
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Henn and Buchter Generally, the probability to have m hits in Kenotype n is: P(m I n) =
(z)
(n lorn) (70) *
The largest single winning probability is reached for 2 hits in Kenotype 3 with P(213) = 0.174. The probability to win anything at all in Kenotype n is given by P(Win 1 Type n) =
2
P(m I n) .
m=O
rn leads to profit
This winning probability takes its maximum for Kenotype 4 with P(Win I Type 4) = 0.321 and its minimum for Kenotype 2 with P(Win I Type 2) = 0.079. As for the “Lottery of Casanova” an interesting task for Keno is to analyze the state’s risk for each of the game options. 7. CONCLUSION This paper on making use of the “Lottery of Casanova” in university teacher education shows some aspects of problem contexts which have a positive effect for productive learning environments. To design a teaching-learning environment according to current conceptions in mathematics education contexts including these aspects have to be developed. Searching for and developing such contexts is therefore essential for the further development of mathematics education. But a good context alone is not sufficient. It has to be used for good teaching, which is not always easy in a concrete teaching-learning situation. It is important that enough time is planned for individual learning and the reflection on this processes, as well as for different solution strategies and their discussion.
REFERENCES Buchter, A. and Henn, H.-W. (2004) Stochastische Modellbildung aus unterschiedlichen Perspektiven - Von der Genueser Lotterie uber Urnenaufgaben zur Keno Lotterie. Stochastik in der Schule, 24 (3), 28-4 1. Childs, J.R. (1961) Casanova: A Biography Based on New Documents. London: George Allen and Unwin Ltd. Freudenthal, H. (1 983) Didactical phenomenology of mathematical structures. Dordrecht: Reidel. Kratz, 0. and Merlin, H. (1995) Casanova. Liebhaber der Wissenschaften. Munchen: Callwey. Wittmann, E.Ch. (1995) Mathematics education as a “design science”. Educational Studies in Mathematics, 29,355-374.