13th International Congress on Mathematical Education Hamburg, 24–31 July 2016 1 - 1 BETWEEN FEAR AND GREED: THE SIX LOOSES Joachim Engel Ludwigsburg University of Education, Germany engel@ph-ludwigsburg.de The paper reflects on students’ intuitive strategies in a game of chance and contrasts their reasoning with a normative probabilistic point of view. The game involves selecting optimal strategies, outweighing potential gains with small probabilities with more probable losses and provides some insight into students’ probabilistic reasoning. INTRODUCTION In the dice game „The Six Looses“ a fair die is being tossed repeatedly and the up face is recorded. As long as no 6 occurs, the rolled numbers are added up to a score. The game can be stopped at any time; then the obtained score is the prize (e.g. as money in Euros) won. However, if a 6 shows up, everything is lost and the game is over. If, for example, the die shows the consecutive numbers 4, 2, 3, 3 and then you stop, the gain is 12 Euro. With the sequence 2, 1, 6 you go empty handed, because you didn’t stop after the second roll and the 6 in the third roll destroys all your previous gains. Imagine you play the game very often. Which strategy for stopping the game should be employed to get on average over many games a score as high as possible? In other words, what is an optimal strategy to maximize your expected gain? We presented this game to college students and inquired about their choice of strategy. While the students were familiar with basic concepts of probability such as the notion of the expected value, they lacked the formal probabilistic knowledge for a full mathematical analysis of this problem. Therefore, they were challenged to reason informally about risks involved in continuing to roll the die, based on intuition and guts feeling. We present and discuss students’ answers followed by a normative presentation of an optimal stopping strategy. THE DICE GAME The game “The Six Looses” was presented to 46 second-year students preparing to be secondary school mathematics teachers. Figure 1 informs about the tasks the students were asked to do. After devising a strategy and noting down a rationale for the proposed approach students were asked to actually play the game at least ten times in a row sticking exactly to their recommended strategy. Thus, they collected some empirical evidence regarding the efficacy of their proposed action. Finally, after having played the game, they were asked to reconsider their originally proposed strategy and reason if they stick to the original plan or favor a possible change of strategy. In previous class meetings students were introduced to the concept of probability and its different notions (classic, frequentist, subjective), learned about computing probabilities in a multi-stage situation via tree diagrams and were introduced to the notion of the expected value as a theoretical average of repeated outcomes that is based on the law of large numbers. Thus, it can be expected that the participating students were capable of calculating the probability of, say, “no 6” occurring in 1, 2, 3, … repeated rolls of a die. Below we present a mathematical