Alford - Calmi - Dang 1

Ring Of Fire

I. Description, Rules & Directions

The “Ring of Fire” is a spinner game with thirty-eight different slots. Nineteen of

those slots are gold, while the other nineteen slots are red. To play the game the player

must insert a marble into the spinner as it spins. Depending on what colored slot the ball

lands in determines if the player shall continue or not. The game ends at one round if

the marble lands in a red slot, but if it lands in a gold slot than the player can continue

until landing two more times in gold or terminating their chance by landing a red.

Rules:

1. No cheating! Be honorable.

2. You have three rounds.

3. The ball must land in a gold space all three rounds for you to win.

4. If the ball lands in a red space on ANY of the three rounds, you automatically

lose. Do not continue.

5. You have 3 seconds after you spin the center of the spinner, to roll your ball. If

you fail to do so, you must re-spin the spinner. This means you may not wait for

the spinner to slow down.

 Alford - Calmi - Dang 2

Directions:

1. Place a coin on the table to play.

2. Round One: Spin the center and roll the ball around the inner edge of the wheel.

3. Wait and check to see where the ball lands.

a. If the ball lands in a gold space, proceed to Round Two.

b. If the ball lands in a red space, you lose. Game over.

4. Round Two: Remove the ball, spin the center and roll the ball again.

5. Wait and check to see where the ball lands.

a. If the ball lands in a gold space, proceed to Round Three.

b. If the ball lands in a red space, you lose. Game over.

6. Round Three: Remove the ball, spin the center and roll the ball again.

7. Wait and check to see where the ball lands.

a. If the ball lands in a gold space, you win! Take three coins!

b. If the ball lands in a red space, you lose. Game over.

 Alford - Calmi - Dang 3

Analysis of Ring Of Fire

II. Theoretical Probability I

Gold Win
Gold Red Loss
Gold
Red Loss

Red Loss
Figure 1. Tree Diagram

Figure 1, shows the possible outcomes when rolling the ball on the spinner. It

shows that if the ball lands on gold, the player proceeds to the next round, while if the

ball lands on red, they lose. This tree diagram helps determine the sample space, which

is stated below.

Sample Space: {GGG, GGR, GR, R}

Figure 2. Sample Space

The sample space is the set of all possible outcomes in an experiment. Gold is

represented as a ‘G’ while Red is represented by an ‘R’. In this case, the sample space

is GGG, GGR, GR, R. This means that the player only has four possible outcomes of

their game. However, this does not mean that each outcome is equally likely. The player

could land on a red on their first roll, land on a gold space and then land on a red space,

land on a two gold spaces then a red space, or land on a gold space all three rounds.

Only one outcome is considered a win, which is GGG.

 Alford - Calmi - Dang 4

P(Win) = P(G) * P(G) * P(G)

P(Win) = 0.5 * 0.5 * 0.5 = 0.1250
Figure 3. Probability Winning

The probability of winning is equal to the probability of getting three golds in a

row. Each turn has a 19/38 chance, or 1/2, chance of landing on gold which results in

the probability of winning being equal to 1/2×1/2×1/2. When calculated this is found to

be 1/8, or 0.125.

1
P(Loss on First Spin)=2
Figure 4. Probability of Loss on First Spin

The probability of a loss on a single spin is 19/38 chance, or 1/2, this means that

the probability of losing on the very first spin is,obviously, 1/2.

P(Loss on Second Spin)=P(First Spin Win)×P(Loss Second Roll)

P(Loss on Second Spin)=1/2×1/2
P(Loss on Second Spin)=1/4
Figure 5. Probability of Loss on Second Spin

As previously stated, the probability of a loss on a single spin is 1/2. In order to

reach the second spin one must win the first spin, which has a probability of 1/2. This in

turn means that the probability of losing on the second spin is 1/2×1/2, or 1/4.

P(Loss on Third Spin)=P(First Spin Win)×P(Win Second Roll)×P(Loss Third Roll)

P(Loss on Third Spin)=1/2×1/2×1/2
P(Loss on Third Spin)=1/8
Figure 6. Probability of Loss on Third Spin

As stated in Figure 5, the probability of losing on the second spin is 1/4, which is

the same as the probability of winning on the second spin. This value can then be
1 1
multiplied by the probability of loss on spin three, 2, which results in a 8 probability of

losing on spin three.

 Alford - Calmi - Dang 5

P(Loss Overall) = 1 - P(Win)

P(Loss Overall) = 1 - 0.1250 = 0.8750
Figure 7. Probability Losing

The probability of losing is equal to 1 minus the probability of winning. After

substitution this becomes 1-0.1250 which when calculated this is found to be 7/8 or

0.8750.

III. Theoretical Probability II

Expected Value = P(Win) * What you win + P(Loss) * What you lose
= 0.1250 * 2 + 0.8750 * (-1)
= -0.625
Figure 8. Theoretical Expected Value

The theoretical expected value is the amount the player is the amount of money

the player is expected to lose in the long run. Found by taking the probability of winning

and multiplying it by the amount of money earned if the player wins. Then, this value is

added to the probability of losing multiplied by the amount of money the player loses if

they do not win. The amount of money earned when the player wins is two chips instead

of three because the amount of money the game costed to play was taken into account.

The theoretical expected value was found to be -0.625, which means in the long run,

the player is expected to lose about 0.625 chips.

IV. Relative Frequency/Experimental Probabilities

Simulation 1: Playing the Game:

 Alford - Calmi - Dang 6

Table 1
Trial Results
Trial Result
1 R
2 GGR
3 GGR
4 GGR
5 GR
6 GGG
7 GR
8 GGR
9 R
10 GGR
11 R
12 GR
13 R
14 R
15 R
16 R
17 GR
18 R
19 GGG
20 R
21 GGG
22 GGR
23 GR
24 R
25 GGR
 Alford - Calmi - Dang 7

Trial Result
26 GR
27 GR
28 GR
29 R
30 GR
31 GGG
32 GR
33 R
34 R
35 R
36 GR
37 GR
38 R
39 GR
40 R
41 R
42 R
43 GR
44 R
45 R
46 R
47 R
48 R
49 GGG
50 GGG

This is the complete table of the results in all 50 trials. Each trial conducted, had

either an outcome of GGG, GGR, GR, or R.

 Alford - Calmi - Dang 8

Table 2
Number of Outcome Occurrences
Outcome Number of
Occurrences

GGG 6

GGR 7

GR 14

R 23

After all of the 50 trials were completed, these values were the amount of times

each outcomes appeared. GGG, which is considered a win, appeared 6 times out of 50

trials. GGR appeared 7 times, GR appeared 14 times, and R appeared 23 times, out of

50 trials. Those were considered losses.

Table 3
Win and Loss Occurrences
Outcome Number of
Occurrences

Win 6

Loss 44

This shows the amount of total wins and losses in all 50 trials. There were 6 wins

and 44 losses in total.

P(Win) = 6/50 = 0.12

P(Loss) = 44/50 = 0.88
Figure 9. Relative Frequency

Relative frequency is the experimental probability. In this experiment or

simulation, there were 6 wins out of 50, making the probability of winning the game

0.12. Which means there were 44 losses out of 50, making the probability of losing the
 Alford - Calmi - Dang 9

game 0.88. This means that 12% of the time the trial was a win, and 88% of the time,

the trial was a loss.

Expected Value = P(Win) * What you win + P(Loss) * What you lose
= 0.12 * 2 + 0.88 * (-1)
= -0.64
Figure 10. Experimental Expected Value

The experimental expected value was found just like the theoretical expected

value in Figure 8, except, the relative frequency probabilities were used in replacement

of the theoretical probabilities. The expected value of this simulation was found to be -

0.64, meaning that in the 50 trials, the player lost about 0.64 chips in the long run, while

playing.

Simulation 2. Simulation:

A Monte Carlo simulation was designed for this portion. Instead of using an

online simulator, the random integer function on the TI-Nspire was used to run the

simulation.

The number one would represent the ball landing on a gold space, and the

number two would represent the ball landing on a red space. One trial consisted of

three numbers, either a one or a two, being generated by the calculator. The random

integer function was used to generate the three numbers for each trial. This was done to

complete 500 trials.

 Alford - Calmi - Dang 10

Table 4
Number of Outcome Occurrences
Outcome Number of
Occurrences

GGG 53

GGR 58

GR 149

R 240

After all of the 500 trials were completed, these values were the amount of times

each outcomes appeared. GGG, which is considered a win, appeared 53 times out of

500 trials. GGR appeared 58 times, GR appeared 149 times, and R appeared 240

times, out of 500 trials. Those were considered losses.

Table 5
Win and Loss Occurrences
Outcome Number of
Occurrences

Win 53

Loss 447

This shows the amount of total wins and losses in all 500 trials. There were 53

wins and 447 losses in total.

P(Win) = 53/500 = 0.11

P(Loss) = 447/500 = 0.89
Figure 11. Relative Frequency

Relative frequency is the experimental probability. In this experiment or

simulation, there were 53 wins out of 500, making the probability of winning the game

0.11. Which means there were 447 losses out of 500, making the probability of losing
 Alford - Calmi - Dang 11

the game 0.89. This means that 11% of the time the trial was a win, and 89% of the

time, the trial was a loss.

Expected Value = P(Win) * What you win + P(Loss) * What you lose
= 0.11 * 2 + 0.89 * (-1)
= -0.68
Figure 12. Experimental Expected Value

The experimental expected value was found just like the theoretical expected

value in Figure 8, except, the relative frequency probabilities were used in replacement

of the theoretical probabilities. The expected value of this simulation was found to be -

0.68, meaning that in the 500 trials, the player lost about 0.68 chips in the long run,

while playing.

Simulation 3. Java Program:

Figure 13. Java Summary

Above is the summary for the Java simulation of 5000 trials. The program itself

looped 5000 times. It would first generate a random number, 1 or 2, and if it generated 1

it would count as a red spot on the spinner and add a value to loss. If it generated a 2 it

would move on repeated this process twice more to simulate three spins of the spinner.

At the very end the program tells the user the results, number of wins, losses on first

spin, losses on second spin, and losses on third spin. It then finishes by calculating the

probability of winning, losing, and the expected value for that simulation run.
 Alford - Calmi - Dang 12

Compare and Contrast Simulations:

When comparing all three trials it appeared that all three were very similar to

each other when it came to the probability of winning or losing. The three simulations

differed in the way they were conducted, Simulation 1 was conducted by doing real

game trials, Simulation 2 was done by creating a Monte Carlo simulation and simulating

it on the TI-Nspire, and lastly Simulation 3 was done by creating a java program.

The theoretical probability of winning was found to be 0.1250. However, the

probability of winning for Simulation 1 was 0.12, Simulation 2 was 0.11, and Simulation

3 was 0.128. All three probabilities of winning were very close to the theoretical

probability, but Simulation 1 was the closest. The theoretical probability of losing would

be the complementary, 0.8750. The experiment probabilities of losing would also be the

complementary of their corresponding probabilities of winning. Again, Simulation 1 was

closest to the theoretical probability.

V. Summary

This game would be great for making money because the expected value of -

0.84 shows that the game is in the operators favor. Which means that the game would

be making a profit over the duration of play time. It also is in the operators favor

because there is only a one-eighth chance of winning due to the three step process of

the game.

The theoretical expected value was found to be -0.625. This means the player

would lose about -0.625 chips in the long run. The expected value for Simulation 1 was

-0.64, Simulation 2 was -0.68 and Simulation 3 was -0.616. Simulation 1 was run for 50
 Alford - Calmi - Dang 13

trials, Simulation 2 was run for 500 trials, and Simulation 3 was run for 5000 trials. This

is brought up because the Law of Large Numbers supports that the larger the number of

trials, the closer the experimental probability or expected value should be to the

theoretical probability or expected value.

Convincing Paragraph:

Mrs. Dewey, you would really want to buy our game for a multitude of different

reasons. First, our game has a one-eighth probability of winning, which means the other

seven-eighths would be chances for you to gain a chip/dollar. Adding onto that, the

player doesn’t just get a three chips/dollars for landing once on a gold, no, they have to

land on gold with the marble three times. Just one time on red and their done and

considering that the spinner is split half and half with the red and gold, well that just

means you have great chances. Furthermore, the expected value is around -0.64, which

means that you would be making a profit over time with this game for the fundraiser.

The numbers from our simulations can also provide backing for all of this, for example

the mean of all three games probabilities of losing is 0.88, which means that the

probability of winning between all three simulations is 0.12. Just proving that this game

would be one that you should invest into buying.

Perspective:

The perspective taken throughout the paper, was the players perspective. For

example, the expected value. The theoretical expected value was found to be -0.625,

because the amount of chips/money lost by the player, if the player was to lose, is 1,
 Alford - Calmi - Dang 14

and the amount of chips/money gained by the player if the player was to win, is 3. This

means the player is expected to lose -0.625 chips/dollars in the long run. If this was

taken in the operators point of view, it would mean the operator was making 0.625

chips/dollars in the long run.

Group Contributions:

Our group consisted of Alex Calmi, Lynn Dang and Katherine Alford. Alex Calmi

contributed with part of Theoretical Probability I, Simulations 2 and 3, Appendix A, and

with some formatting and anchoring. Lynn Dang contributed with the rules, directions,

part of Theoretical Probability I, Theoretical Probability II, Simulation 1, getting the

supplies and decorating the spinner, parts of the compare and contrast, parts of the

summary, and some formatting and anchoring. Katherine Alford contributed by the

description, parts of the compare and contrast, parts of the summary, and peer-editing.
 Alford - Calmi - Dang 15

Appendix A: Java Program Simulation