Ciaramitaro Dobranski Weidemann

10C March 21st, 2017

Trumped Up Carnival

Have you ever dreamed of a carnival game that was not only fun to play, but also a great

tool for getting students interested into the history of US presidents? Imagine a simple game in

which no skill was involved, all that had to be done was roll a die, and random probability would

determine the outcome. There would be fantastic prizes, ranging from candy suckers to miniature

American flags… perhaps even a sticker of the current US president! This amazing contraption

does exist, and the following will explain how to play, and your chances at winning.

Description:

A presidential themed carnival game... what better way is there to get the youth of

America interested in politics? Although featuring the randomness element of dice, this carnival

game incorporates an extra mixup of probability with a drop tube and designated “prize” spots at

the bottom. To better suit the theme, the standard six number die has been replaced with a

patriotic, star spangled, red; white; and blue probability cube. Prize spots are labeled with images

of some of our country's former (and current) presidents. Classics such as dum-dum suckers,

miniature American flags, and even presidential stickers are the potential prizes. This simple yet

addictive game can provide endless fun, all for the low price of 50¢.

Rules:
1. Do not look down the center tube when dropping the die.
2. Do not attempt to move the box while the die is being dropped.

Directions:
1. Carefully pull the lid off the game and take the die out of the box.
2. Place the lid back on the box, ensure that the lid is properly secured, and lies flat on the
top of the box.
3. Take the die and drop it down the tube.
 Ciaramitaro Dobranski Weidemann
10C March 21st, 2017

4. Open the box to observe if a prize has been won or not. The criteria for winning a prize is
as followed:
a. The die lands on a picture of Donald Trump, and has a faceup value of red:
→ Donald Trump Sticker
b. The die lands on a picture of Barack Obama:
→ Miniature American flag
c. The die lands with a faceup value of red:
→ Win a dum-dum sucker
(If multiple sets of criteria were met on your attempt, take the highest level prize,)
5. Leave the die in the box for the next player.

Figure 1. Carnival Game Diagram

The above figure shows a diagram for the design for this game. The box with the four

pictures is at the top while the lid with the tube is on the bottom. The lid will be originally on the

top of the box before the game is played. Once the player takes the lid off the player must replace

it after their turn is completed.

Table 1
Possible Outcomes
 Ciaramitaro Dobranski Weidemann
10C March 21st, 2017

Trump Obama Reagan Kennedy

Red 1/12 1/12 1/12 1/12

White 1/12 1/12 1/12 1/12

Blue 1/12 1/12 1/12 1/12

Table 1 shows all of the outcomes that are possible in each trial. These outcomes make

up the full sample space and add up to one. Every outcome has the same probability, which

means each outcome is as likely as the next to be rolled when someone is playing the game.

Table 2
Prize Possibilities
Prize Level Needed Outcome Probability of Winning

Top P(Die: Red ∩ President: Trump) 1/12

Secondary P(President: Obama ∩ 𝑅𝑅𝑅) 2/12

Tertiary P(Die: Red) 3/12

Overall:
6/12

Table 2 shows the possibilities of winning a prize. The top prize, which is rolling a red on

the Donald Trump picture, has a probability of 1/12. The second place prize, which is rolling the

dice (any color besides red) onto the picture of Barack Obama, has a probability of 2/12. The last

prize is won when the die lands on red. This has a probability of 3/12. If the die were to roll a red

and land on Barack Obama, then the third place prize will be awarded as the red correlates to the

republican party which is now in office. The probability of losing would be the complement of
 Ciaramitaro Dobranski Weidemann
10C March 21st, 2017

the total overall chance of winning, or 6/12. The overall perspective for this paper will be in

regards to the player which will be a loss.

Table 3.
Probability Distribution Chart
$ -0.50 -0.35 -0.20 +2.50

P($) 6/12 3/12 2/12 1/12

Table 3 above shows a probability distribution chart for the possible outcomes of the

game. In order of left to right, the prize cost and probability of winning goes from a loss, to a

tertiary prize win, to a secondary prize win, all the way to a top level prize. It lists the monetary

value of the prizes when the cost to play ($0.50) is taking into account, alongside the probability

(found in table 2) of them occurring. This means that the top level prize costs $3.00, after

subtracting the initial cost, you’re left with a “profit” of $2.50. These values can then be plugged

into the expected value equation (as shown in the next figure) which will then provide the overall

expected value (which will be shown in Figure 2).

𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅𝑅
= −0.5(𝑅) − 0.35(𝑅) − 0.20(𝑅)
+ 2.5(𝑅)
Figure 2. Expected Value Formula

In Figure 2, the formula for finding the expected value of a randomized game is shown.

Each variable represents the probability of winning a certain prize (n for no prize, t for tertiary

prize, s for second prize, and f for first/top prize). The value those variables are multiplied by are

the monetary gain/loss the player receives upon earning those values.
 Ciaramitaro Dobranski Weidemann
10C March 21st, 2017

𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅𝑅
= −0.5(6/12) − 0.35(3/12) − 0.20(2/12) + 2.5(1/12)
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑅𝑅𝑅𝑅𝑅 = −0.1625
Figure 3. Expected Value

The above figure shows how the expected value was calculated. Each monetary value and

its corresponding probability were multiplied together. Each of those answers were then added to

get the expected value of -0.1625. This means that, in the long run, the player will lose about 16

cents every time they play this game.

Table 4
Results of 50 Trial Simulation
Trial Results Win?

1 WR No

2 WR No

3 RK 3rd

4 BR No

5 WK No

... ... ....

45 WT No

46 BT No

47 WK No

48 WK No

49 RK 3rd

50 RO 3rd

Table 4 shows the relative frequencies of the first and the last 5 trials in the 50 trial

simulation. The data that was taken includes the outcome of each roll and if that roll won. The
 Ciaramitaro Dobranski Weidemann
10C March 21st, 2017

table also shows the prize that was won in each trial. This data was collected by manually

playing the game 50 times.

Table 5
Contingency Table of 50 Trial Simulation
Trump Obama Reagan Kennedy Total

Red 5 2 2 5 14

White 5 5 6 11 27

Blue 2 5 2 0 9

Total 12 12 10 16 50

Table 5 above shows the results in a contingency table of the 50 trials ran. Most of the

totals for each president seem just about spread throughout indicating no clear bias. The totals of

the colors on the die showed some bias toward the color white which indicates the dice could be

unfair. Another reason for this might be because of the law of large numbers and that there might

not have been enough trials to bring these results closer to the theoretical probability.

Total Loss of Player = (5 × 2.5) + (10 × (−0.2)) + (9 × (−0.35)) + (26 × (−0.5))

Figure 4 shows that the player lost $5.65. This was calculated by multiplying the

monetary value of each prize to its corresponding relative frequency. On average, the player lost

about 11 cents per trial. This was calculated by dividing the total loss by the number of trials

which in this case is 50. The expected value was calculated to be -0.1625 which is somewhat
 Ciaramitaro Dobranski Weidemann
10C March 21st, 2017

close to the average loss of -0.113 from this set of results. This number is not exact because of

the law of large numbers. The more trials runned allows the average loss to get closer and closer

to its expected value. Fifty trials was not enough to get the average loss to its expected value.

Figure 5. 500 Trial Simulation

Figure 5 is a screenshot taken, from random.org, of 500 numbers generated randomly 1-

12. These numbers were generated by random.org to insure randomness of results. Simulation

two was designed by having random numbers generated 1-12 500 times from random.org. When

a number was 1-6, it was marked as a loss. If a number was 7-9, then it was marked as a winner

of a third place prize. If a number was 10 or 11, then it was marked as a second place prize.

Lastly, if the number were to be a 12, then it was marked as the main prize. One of these

numbers generated is counted as a trial so there is a total of 500 trials.The amount of each

number was counted and put into a contingency table as shown in the next table.
 Ciaramitaro Dobranski Weidemann
10C March 21st, 2017

Table 6
Contingency Table of 500 Trial Results
Number 1-6 7-9 10-11 12
Generated

Relative 258 122 76 44

The above table shows the results of a simulation of 500 randomly generated numbers.

The numbers are categorized into their corresponding prizes. Out of these 500 trials, the player

of the game is calculated to lose about $77. This number was calculated by multiplying the

number of times each outcome occurred to its corresponding monetary profit, then adding all of

products up. The average amount of money that the player lost per trial is $-0.1538. This means

that on average the player will lose 15 cents every trial. This was calculated by dividing the total

amount lost by the player and dividing it by the number of trials which in this case will be 500.

Figure 6. Java Program Results

In Figure 6, the results given from the java program simulation are shown. These results

accurately line up with the expected values. Top prize winners account for roughly a twelfth of

the total simulated game's, second prize winners are nearly double that, and the tertiary prize

winners are just under triple. The expected value ( or average loss) indicates that everytime the

game is played, the player loses about $0.16 . This simulation was done by generating two

random numbers, the first ranging from one to four (representing the president the die may land
 Ciaramitaro Dobranski Weidemann
10C March 21st, 2017

on), and the second ranging from one to three (representing the faceup color on the die). From

there, a series of checks ran to see if they matched up with the necessary criteria to win the game,

and they were stored each time a match occurred. The average loss was determined with the

math equation seen in figure 2 with variable being replaced by the stored winner amounts. To see

the full java program that was created and the equation used to calculate the average loss, see the

“Appendix” at the end of the paper.

These simulations all share similar results, they are relatively close to the theoretical

probability. As the amount of trials played increases, the average loss becomes closer to what it

is to be expected as shown in Figure 2, otherwise being the law of large numbers.

Summary

Steve Jobs is quoted with saying “Great things in business are never done by one person.

They're done by a team of people.” . Trumped Up Carnival is an excellent example of this, a

collaborative group effort to produce the greatest presidential themed carnival game to have ever

been analyzed. From the business side, not only is it never ending fun for people of all ages, it is

also quick to turn a profit for the operator with a respectable average loss for the player of just

over 16¢ per game. If five hundred people where to play this game, that would create a profit of

exactly $80, which could then be used to purchase a few dozens brownies to hand out at the next

fundraiser. Alternatively, you could set up one of these contraptions at every voting booth in

America. If every single person who voted in the 2012 presidential election played this game, a

profit of over $20,653,664.48 would be made!

An analysis of the theoretical probability involved showed great results for both the

player and the operator. Half of all games played should result in some sort of prize given out,
 Ciaramitaro Dobranski Weidemann
10C March 21st, 2017

though due to their relatively low costs the operator is still able to make a profit of 16.25¢ on

average. Upon actual trials being performed, the outlook seems to be dark and gloomy as more

are performed. At fifty trials, the results can go either way to do the relatively small sample

space, it is not horribly unlikely for a player profit to actually be made. However, towards five-

hundred and five-thousand trials, the law of large numbers becomes clearly present and the odds

of player returning a profit becomes slim to none as the relative frequency approaches the

theoretical. This was seen as the results with a fewer amount of trials produced an expected value

of -0.113, as more trials were added, this value got closer and closer to what it should be. With

five-hundred trials, the value was -0.1538 and at five thousand it was -0.1626. Of course, is

fantastic news for any entrepreneur looking for a money making machine.

Back to Steve Job’s quote, this carnival game was a group effort, and could not have been

accomplished without the work of everyone involved. The entirety of the project was divided up

into three separate parts. Those being the construction, the math and logistics, and the writing,

organizing, and formatting the paper itself. While everyone did partake in various aspects of the

project, certain people took leads on what they were believed to be best at, in order to produce

the best possible result. Andy was the main person to work with the construction, and making

sure that all the dimensions, parts, and design tied together nicely. Nathan paved the way for the

paper, making sure that all aspects were incorporated and requirements met, alongside setting up

the initial java program. Reis was responsible for reasoning out how the game would work, what

conditions needed to be met for certain scenarios to occur, and getting the probabilities of each

respective event occurring. This evenly distributed the workload, and was extremely efficient in

getting the task accomplished.

Appendix