FACES
A Casino Game
BERGES IRANI
1
TABLE OF CONTENTS
Rules ..............................................................................................................................................................................2
Setup ..........................................................................................................................................................................2
Mathematical content ...................................................................................................................................................3
Theoretical Probability Graphs and charts ................................................................................................................3
Theoretical Probability Analysis .................................................................................................................................5
Organized Experimental Outcomes ...........................................................................................................................6
Experimental Probability and Graph .......................................................................................................................11
Data Analysis............................................................................................................................................................11
Conclusions ..................................................................................................................................................................12
Problems That Arose ...............................................................................................................................................12
Summary of Games Fair...........................................................................................................................................12
Future Modifications ...............................................................................................................................................12
Score ............................................................................................................................................................................12
2
RULES
1. Pick 6 cards from a deck.
2. Count the amount of face cards you have chosen
3. Role the die once
4. If die shows the number of face cards you have, you win!!
5. Calculate your reward using the table below
# Faces
0
1
2
3
4
5
6
Reward($)
0
2
4
6
8
10
12
SETUP
The game is simple to setup. First assure that the text file named test is in the right folder, the game folder. Then
make sure that the test file is empty. Run the game by double clicking the icon. The game will turn on and prompt
the player to play. When the experiment is over, hit the x button in the top right corner to shut down the game
and save all trials to the text file test.
3
MATHEMATICAL CONTENT
THEORETICAL PROBABILITY GRAPHS AND CHARTS
Number of Face Cards Dice rolls Points earned/lost Probability
xP(x)
Sum of xP(x) Probability of winning
1
0 0.031423107
0 0.461538372 0.135243
2
0 0.031423107
0
Probability of losing
3
0 0.031423107
0
0.864756
4
0 0.031423107
0
5
0 0.031423107
0
0
6
0 0.031423107
0
1
2 0.064642027 0.129284054
2
0 0.064642027
0
3
0 0.064642027
0
4
0 0.064642027
0
5
0 0.064642027
0
1
6
0 0.064642027
0
1
0 0.049379326
0
2
4 0.049379326 0.197517304
3
0 0.049379326
0
4
0 0.049379326
0
5
0 0.049379326
0
2
6
0 0.049379326
0
1
0 0.017794351
0
2
0 0.017794351
0
3
6 0.017794351 0.106766106
4
0 0.017794351
0
5
0 0.017794351
0
3
6
0 0.017794351
0
1
0
0.00316083
0
2
0
0.00316083
0
3
0
0.00316083
0
4
8
0.00316083 0.02528664
5
0
0.00316083
0
4
6
0
0.00316083
0
1
0
0.00025935
0
2
0
0.00025935
0
3
0
0.00025935
0
4
0
0.00025935
0
5
10
0.00025935
0.0025935
5
6
0
0.00025935
0
1
0 0.000007564
0
2
0 0.000007564
0
3
0 0.000007564
0
4
0 0.000007564
0
5
0 0.000007564
0
6
6
12 0.000007564 0.000090768
4
PROBABILITY OF WINNING BASED ON NUMBER
OF FACE CARDS PICKED.
# FACES = 2
# FACES = 3
0.999992436
0.000007564
0.99974065
0.00025935
0.00316083
0.017794351
0.049379326
# FACES = 1
0.99683917
0.950620674
0.935357973
0.064642027
1
0
# FACES = 0
Losing
0.982205649
Winning
# FACES = 4
# FACES = 5
# FACES = 6
Reward Probability
1
0.864756552
0.9
0.8
Probability
0.7
0.6
0.5
0.4
0.3
0.2
0.064642027
0.1
0.049379326
0.017794351
0.00316083
0.00025935
0.000007564
6
8
10
12
0
0
2
4
Reward($)
14
5
Reward Probability not including 0
0.07
0.064642027
0.06
0.049379326
Probability
0.05
0.04
0.03
0.017794351
0.02
0.01
0.00316083
0.00025935
0.000007564
10
12
0
0
2
4
6
8
14
Reward($)
THEORETICAL PROBABILITY ANALYSIS
The theoretical probability calculations how that the sum of xP(x) is 0.461538372. This means the expected
winnings is approximately 46 cents. This is good for the game players because they only pay $1 to play. This is even
better for the casino because they are expected to make 54 cents for every game played. The data also shows that
it is highly improbable that a large sum of money can be won, such as 10 or 12 dollars. It is also evident that
smaller amounts of money have the highest probability of being won. This makes the game profitable but also
allows for frequent and rewarding give-aways.
6
ORGANIZED EXPERIMENTAL OUTCOMES
# Faces Dice
2
4
2
1
4
2
0
0
2
2
0
2
3
1
3
0
1
2
1
2
1
1
2
1
1
2
4
1
1
1
1
1
1
2
2
1
1
1
0
2
Win/Lose Reward
1 L
0
3
5
3
5
5
1
4
1
2
4
1
2
3
3
5
5
5
3
6
3
3
4
5
6
1
1
6
2
2
1
5
2
5
1
4
5
3
4
1
L
L
L
L
L
L
L
L
W
L
L
L
L
W
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
W
L
L
L
L
L
L
L
L
L
0
0
0
0
0
0
0
0
4
0
0
0
0
6
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
Total cost =
Gross income + float
=
Net Income =
56
267
211
7
3
1
1
2
4
2
0
4
0
1
1
1
0
1
3
3
2
1
1
2
3
1
2
1
2
1
2
2
2
1
2
2
2
1
0
0
3
3
2
2
3
0
1
1
1
5
2
4
3
5
5
1
1
3
4
4
2
1
5
2
6
4
6
3
4
1
6
4
6
3
4
5
2
5
1
5
3
5
1
3
5
3
2
3
3
2
5
5
L
L
L
L
L
L
L
L
L
L
L
L
L
W
L
L
L
L
L
L
L
W
L
L
L
L
L
L
W
L
L
L
L
L
L
L
L
W
W
L
W
L
L
L
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
2
0
0
0
0
0
0
4
0
0
0
0
0
0
0
0
6
4
0
6
0
0
0
8
0
2
1
3
0
3
2
0
0
2
1
0
1
1
0
3
2
1
3
2
1
1
1
2
2
0
1
2
1
0
1
2
0
1
1
1
2
2
1
2
0
1
1
1
4
2
5
3
4
2
5
6
3
5
3
3
6
6
5
5
6
1
2
6
5
3
2
6
6
3
6
1
5
6
6
1
3
1
3
2
1
4
1
1
5
3
4
6
L
W
L
W
L
L
L
L
L
L
L
L
L
L
L
L
L
W
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
W
L
L
L
L
W
L
L
L
L
L
0
4
0
6
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
2
0
0
0
0
0
9
3
3
2
1
1
0
1
1
0
2
2
1
0
2
3
1
0
3
1
1
1
1
1
2
2
2
1
2
2
0
0
2
2
0
1
2
4
0
1
4
4
6
2
6
1
2
1
3
3
1
6
2
3
2
3
3
6
6
5
4
1
4
6
5
2
6
3
5
5
5
4
1
3
5
4
1
2
3
L
L
L
L
L
L
L
W
L
L
L
L
L
L
L
L
L
L
L
L
L
W
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
10
GAMES WON AND LOST
Won, 16, 10%
Lost, 151, 90%
Rewards Won ($) Not Including $0
0, 0%
0, 0%
0, 0%
4, 25%
2
4
8, 50%
6
8
10
12
4, 25%
11
EXPERIMENTAL PROBABILITY AND GRAPH
DATA ANALYSIS
The theoretical probability for Faces is very close to the outcomes of the game. This is evident in the theoretical
expected winnings and the experimental expected winnings. Theoretically, the expected winnings are 46 cents per
game, compared to the experimental expected winnings which is 36 cents. Only a ten cent difference means that
the experimental and theoretical expected winnings are relatively close to one another. In addition to the
expected winnings, the experimental probability of winning and losing is extremely close to that of the theoretical
probability. There was a mild difference in the fact that the experimental probability showed no winnings greater
than 6 dollars unlike the theoretical probability. This is why the game made a total of 211 because the probability
of winning a large amount of money is very low, the experiment matches the theoretical probability.
12
CONCLUSIONS
PROBLEMS THAT AROSE
After a few games, an issue of the program itself arose. It would stop working and players could no longer press
the same bar to pick cards or roll the dice. This made it frustrating for many eager gamers, it caused long lines to
play the game, as well as a delay in trials. It became clear later on that the game itself was not broken but that the
players would click outside of the game window, stopping player input (space bar) from being read. The touchpad
on the computer was turned off so that players cannot click outside the game window
The games integrity was questioned. The game is a computer program, not something physical that the players can
feel and touch. This caused players to accuse the game of being rigged. After losing money, many players lost hope
and came to the conclusion that the game is a con and will never actually allow you to win any prizes. This issue
addressed itself. When a few people won large prizes that multiplied their investment by 6, the integrity of the
game was restored. More and more players came to play the game as it gave out large winnings.
A consequence of the game being simple and quick to play, many players bought multiple tickets in advance (up to
20). This caused a major issue because other players did not get the chance to participate. When the player who
paid 20 dollars in advanced played, he placed a chair in front of the game and spent a very long time playing. This
frustrated the players behind him in the line. To fix this issue, a limit on how many games you can play at once was
used to stop this type of issue. This allowed for everyone to play a fair amount of time.
SUMMARY OF GAMES FAI R
The games fair went well. A total of 168 people played Faces, this can be viewed as a success because it includes
each person in the class playing multiple times. The game also earned a lot of positive feedback, such as the fact
that it was highly addictive. Many players had a hard time leaving the game because they knew they could make a
lot of money, and the game was very simple to play. This is another key point about the game that made it very
successful, it was simple. There was no need to think, plan, or strategize, the player must simply hit the spacebar
and be prepared to either win or lose. A negative aspect about the game is the range of winnings, it is very difficult
to win the $12 jackpot. In addition to its unlikelihood, the amount needed to play enough games to finally win the
jackpot is much greater than the prize itself. This made players unhappy because they often felt like they were
losing money, which in some cases is true and in others is not.
FUTURE MODIFICATIONS
If this game were to be played again, it would be wise to include sound effects. Specifically one for when you win,
this would primarily let other players know that somebody had just won and that they can as well. It will allow
players to experience a form of glory, making them want to play the game more. As well as make the game look
much better in comparison to others. The second modification is a list in the corner of the game with all the
winners as well as their prizes. This would be done to demonstrate to players that the game is winnable and that
there are fortunes waiting for them to win. Finally, the jackpot should be increased to entice players to play more
and maybe win more money, this would still be profitable because the jackpot is nearly impossible to win. Evident
in the experiment when no player had won it.
SCORE
((6x16) + 15)/7 = 15.86 = 99%
13
14
15
16