Exercise 1.3Z: Winning with Roulette?

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Considered betting situation

In roulette, a winning number  $Z$  is determined in each game by means of a ball and a roulette wheel, where we want to assume that all possible numbers  $Z \in \{0, 1, 2, \ \text{...} \ , 36 \}$  are equally probable.

The players can now bet on a single number or on a group of numbers with chips of different value.  Some of the possibilities and the corresponding winnings will be briefly explained here on the basis of the chips bet by a player (see graph):

• If a player bets on a number (in the example on "0"), he would get back  $35$ times his stake as winnings.

• If a player bets on a group of numbers with three fields (in the example, the 1-euro chip for the numbers from "22" to "24"), he would receive  $ 11$ times his stake as winnings in addition to his bet.

• If a player bets on a group of numbers with  $ 18$  fields (for example, the 10-euro chips on "Rouge", on "Impair" and on "Passe"), he will receive the same amount back as winnings in addition to his bet.
• If the number drawn does not belong to one of the squares he occupies, his bet is lost.

Hints:

• The exercise belongs to the chapter  Set Theory Basics.

• Enter any losses as negative winnings in the following questions.
• The topic of this chapter is illustrated with examples in the (German language) learning video Mengentheoretische Begriffe und Gesetzmäßigkeiten $\Rightarrow$ Set Theoretical Concepts and Laws.

Questions

1

A player simultaneously places one 1-euro chip on each of the squares „0“, „Red“ und „Black“.  What are his average winnings per game?

$G_1 \ =\ $ $\ \rm Euro$

2

How much does he win on average per game if he always places one  $1$  Euro chip on each of the squares "Red" and "Black"?

$G_2 \ =\ $ $\ \rm Euro$

3

How much does he win on average per game if he always bets  $1$  on "0" and  $10$  Euro auf on "Red"?

$G_3 \ =\ $ $\ \rm Euro$

4

The player bets as shown in the picture.   Which number  $Z_{\rm Wunsch}$  (Wunsch being German for wish/desire) should he be hoping for?  How big would his winnings be then?

$Z_{\rm Wunsch} \ = \ $

$G_4 \ =\ $ $\ \rm Euro$

5

Is there a betting combination such that the average winnings are positive?

	Yes   ⇒   Quit university and go to the next casino.
	No   ⇒   Continue with $\rm LNTwww$.

Solution

Solution

(1)  The player loses one euro each time one of the numbers  $1$  to  $36$  is drawn.

• He wins  $33$  euro, if  $0$  is drawn. It follows that:

$$G_1 =\rm {36}/{37}\cdot (-1\hspace{0.1cm} Euro) + {1}/{37}\cdot (33\hspace{0.1cm} Euro) \hspace{0.15cm}\underline {= - 0.081\hspace{0.1cm} Euro\hspace{0.1cm}(Loss)}.$$

(2)  The player wins and loses nothing unless the zero is drawn.  If the zero appears, he loses his bet:

$$G_2 = \rm {1}/{37}\cdot (-2\hspace{0.1cm} Euro)\hspace{0.15cm}\underline { = -0.054 \hspace{0.1cm}Euro \hspace{0.1cm}(Loss)}.$$

(3)  If "red" is drawn, he wins nine euro.

• If zero comes, he effectively wins  $25$  euro.
• If "black" is drawn, he loses his entire bet of  $11$  euro:

$$G_3 = \rm {18}/{37}\cdot (10 -1) + {1}/{37}\cdot (35-10) + {18}/{37}\cdot (-10-1)\hspace{0.15cm}\underline { = - 0.297\hspace{0.1cm}Euro}.$$

(4)  He gets the highest winning at  $Z_{\rm Wunsch} \; \underline{ = 23} $.  Then four of his five chips win:

$$G_4 = \rm 10\hspace{0.1cm}(Red) + 10\hspace{0.1cm}(Passe) + 10\hspace{0.1cm}(Impair) + \rm 11\hspace{0.1cm}(between \hspace{0.1cm}22\hspace{0.1cm} and \hspace{0.1cm}24) - 1 \hspace{0.1cm}(not \hspace{0.1cm}0) \hspace{0.15cm}\underline {= 40 \hspace{0.1cm}Euro}.$$

• If, on the other hand, the zero comes, he wins only  $\rm 35 - 31 = 4 \ Euro$.

(5)  No, unfortunately not. On statistical average, the house always wins..