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 in addition to his bet.

• 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"  and  "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  $1$  Euro on  "0"  and  $10$  Euro on  "Red"?

$G_3 \ =\ $ $\ \rm euro$

4

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

$Z_{\rm desire} \ = \ $

$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 if 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 desire} \; \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.