Teil B - Wahlaufgaben Wahlaufgabe 6.1 Ein Schausteller bietet das Spiel „Würfelbude' an. WÜRFELBUDE Pro Versuch wirft ein Spieler drei unterscheidbare Würfel. Für jede Augenzahl 6 Bei jedem Würfel ist das Werfen der Augenzahlen 1 bis 6 2 € Auszahlung! gleichwahrscheinlich. Es interessiert jeweils nur, ob die Augenzahl 6 oben liegt oder nicht. a) Zeichnen Sie für dieses dreistufige Zufallsexperiment ein Baumdiagramm. Tragen Sie die entsprechenden Wahrscheinlichkeiten ein. b) Die Zufallsgröße X ordnet jedem Ergebnis des Zufallsexperimentes den Auszahlungsbetrag in Euro zu (siehe Abbildung). Geben Sie an, welche Werte die Zufallsgröße X annehmen kann. Berechnen Sie die Wahrscheinlichkeit für jeden Wert der Zufallsgröße X. Jeder Versuch kostet 1,50 €. Ermitteln Sie, wie viel Gewinn der Schausteller durchschnittlich pro Versuch erwarten kann. Für Wahlaufgabe 6.1 erreichbare BE: 8

Question a)

1. Determine the probabilities for each outcome of a single die roll
   The probability of rolling a 6 is $$\frac{1}{6}$$$$\frac{1}{6}$$, and the probability of not rolling a 6 is $$\frac{5}{6}$$$$\frac{5}{6}$$

2. Draw the tree diagram for the three-stage experiment
   The tree diagram has three stages, each representing a die roll. Each stage has two branches: one for rolling a 6 (probability $$\frac{1}{6}$$$$\frac{1}{6}$$) and one for not rolling a 6 (probability $$\frac{5}{6}$$$$\frac{5}{6}$$)

3. Label the branches with the corresponding probabilities
   The first stage starts with the first die roll. The second stage branches out from the first, representing the second die roll. The third stage branches out from the second, representing the third die roll. Each branch is labeled with either $$\frac{1}{6}$$$$\frac{1}{6}$$ or $$\frac{5}{6}$$$$\frac{5}{6}$$ depending on whether it represents rolling a 6 or not.

The answer is: See explanation

Question b)

1. Determine the possible values of the random variable X
   The random variable X represents the payout amount in Euro. Since each 6 rolled results in a 2 € payout, the possible values for X are 0 €, 2 €, 4 €, and 6 €.

2. Calculate the probability for each value of X

   • P(X = 0 €): This occurs when no 6 is rolled in any of the three dice rolls. The probability is $$(\frac{5}{6})^{3} = \frac{125}{216}$$$$(\frac{5}{6}{)}^3=\frac{125}{216}$$
   • P(X = 2 €): This occurs when exactly one 6 is rolled. There are three ways this can happen (6xx, x6x, xx6, where x represents not a 6). The probability is $$3 \times (\frac{1}{6}) \times (\frac{5}{6})^{2} = 3 \times \frac{25}{216} = \frac{75}{216}$$$$3\times (\frac{1}{6})\times (\frac{5}{6}{)}^2=3\times \frac{25}{216}=\frac{75}{216}$$
   • P(X = 4 €): This occurs when exactly two 6s are rolled. There are three ways this can happen (66x, 6x6, x66). The probability is $$3 \times (\frac{1}{6})^{2} \times (\frac{5}{6}) = 3 \times \frac{5}{216} = \frac{15}{216}$$$$3\times (\frac{1}{6}{)}^2\times (\frac{5}{6})=3\times \frac{5}{216}=\frac{15}{216}$$
   • P(X = 6 €): This occurs when all three dice rolls are 6s. The probability is $$(\frac{1}{6})^{3} = \frac{1}{216}$$$$(\frac{1}{6}{)}^3=\frac{1}{216}$$

3. Calculate the expected payout per attempt
   The expected payout is $$E(X) = 0 \times \frac{125}{216} + 2 \times \frac{75}{216} + 4 \times \frac{15}{216} + 6 \times \frac{1}{216} = \frac{0 + 150 + 60 + 6}{216} = \frac{216}{216} = 1$$$$E(X)=0\times \frac{125}{216}+2\times \frac{75}{216}+4\times \frac{15}{216}+6\times \frac{1}{216}=\frac{0+150+60+6}{216}=\frac{216}{216}=1$$ €.

4. Calculate the average profit for the showman per attempt
   Each attempt costs 1.50 €, and the expected payout is 1 €. Therefore, the showman's average profit per attempt is 1.50 € - 1 € = 0.50 €.

The answer is: 0.50 €