Overview
Probability is the likelihood of something happening in the future. It can be expressed as a number between zero and one. Zero is used to signify that the event in question can never happen and one expresses that the event will always happen. Probability is a very important part of our daily lives. For example, we check the weather forecast every day, which is essentially a prediction of conditions based on previous data. Betting on teams in a sport obviously involves probability because the spectators predict the odds of one of the teams winning or losing.
In class, we also learned that there are many types of probability, namely Observed Probability, Theoretical Probability, and Conditional Probability. Observed probability is the probability obtained by taking measurements. It is essentially the actual probability of an event. Theoretical probability is the probability dictated by the nature of the event, or in other words, the predicted probability. Conditional probability is the probability of an event given that another event has already occurred. In the activity "Dog Ate My Homework," we learned more about conditional probability and expected values. Expected values are the number of times you would expect to get a specific outcome if you repeated an experiment a "very large" amount of times. The activity mentioned above involved a flowchart labeled with certain events that could take place depending on which number on a six-sided die is rolled.
The background information for "Dog Ate My Homework" stated that Dr. Drew claims that he could tell when a student is lying about homework, and most of the time, he is correct in his assumption. A few times, he unfortunately accuses students who are telling the truth. The flowchart starts with rolling a die. If a six is rolled, then the student is lying and gets accused. If not, then the student is not lying. The die is then rolled again and this time, if a one is rolled, the student is accused and if a one is NOT rolled, then the student is believed. We repeated this experiment 36 times and subsequently entered the resulting data into tree diagram and a two-way table. A tree diagram is a visual representation of all the possible outcomes of an event and allows us to calculate their probability. Each branch represents a possible outcome. A two-way table is a statistical table that shows the observed number or frequency for two variables.
In class, we also learned that there are many types of probability, namely Observed Probability, Theoretical Probability, and Conditional Probability. Observed probability is the probability obtained by taking measurements. It is essentially the actual probability of an event. Theoretical probability is the probability dictated by the nature of the event, or in other words, the predicted probability. Conditional probability is the probability of an event given that another event has already occurred. In the activity "Dog Ate My Homework," we learned more about conditional probability and expected values. Expected values are the number of times you would expect to get a specific outcome if you repeated an experiment a "very large" amount of times. The activity mentioned above involved a flowchart labeled with certain events that could take place depending on which number on a six-sided die is rolled.
The background information for "Dog Ate My Homework" stated that Dr. Drew claims that he could tell when a student is lying about homework, and most of the time, he is correct in his assumption. A few times, he unfortunately accuses students who are telling the truth. The flowchart starts with rolling a die. If a six is rolled, then the student is lying and gets accused. If not, then the student is not lying. The die is then rolled again and this time, if a one is rolled, the student is accused and if a one is NOT rolled, then the student is believed. We repeated this experiment 36 times and subsequently entered the resulting data into tree diagram and a two-way table. A tree diagram is a visual representation of all the possible outcomes of an event and allows us to calculate their probability. Each branch represents a possible outcome. A two-way table is a statistical table that shows the observed number or frequency for two variables.
Hazard
The dice game Hazard was supposedly invented in 14th century Europe, although some sources argue that it has Middle Eastern origins because of its roots from the Arabic word for die (al-zahr). The earliest record of Hazard can be found in Geoffrey Chaucer's Canterbury Tales. It was very popular in the 17th and 18th centuries and is a precursor to the game craps which is still popular worldwide today. Hazard was played by everyone from peasants to nobles and everywhere from street corners to lavish feasts. At the height of its popularity, it was not uncommon to play for money, but it is just as interesting without having to gamble. As mentioned previously, Hazard is a precursor to the dice game Craps, which is popular today. Craps was created by a simplification of Hazard's rules.
I did not adapt this game in any way because I felt that the original rules of Hazard were simple enough to be understood easily. Chance is a crucial part of this game because there is no skill involved in winning it whatsoever. Each step of the game is governed by chance in that the way in which you proceed with the game is determined by what number the dice roll. In fact, the only part of the game that is in the player's control is in choosing the main.
I did not adapt this game in any way because I felt that the original rules of Hazard were simple enough to be understood easily. Chance is a crucial part of this game because there is no skill involved in winning it whatsoever. Each step of the game is governed by chance in that the way in which you proceed with the game is determined by what number the dice roll. In fact, the only part of the game that is in the player's control is in choosing the main.
Rules of the Game
Number of players: Any number of people can play, but only one person has the dice at any given time. This person is called the caster.
1. The caster chooses a number between 5 and 9. This number is now called the main.
2. The caster now rolls the dice. If it's the main, they win. If it's a 2 or 3, they lose. The results for 11 or 12 are listed below.
a. If they have a main of 5 or 9, they will lose with 11 or 12.
b. If their main is 6 or 8, they lose with 11 and win with 12.
c. If they have 7 as a main, they will win with 11 and lose with 12.
3. If the caster neither loses nor wins, the number thrown is called the chance. They throw the dice again and one of these three things will happen:
a. If they roll the chance again, they win.
b. If they roll the main, they lose.
c. If they roll neither, then they keep rolling until they get one or the other.
4. The caster remains the caster until they lose three times in succession, after which they pass the dice to the player to their left, who then becomes the next caster.
1. The caster chooses a number between 5 and 9. This number is now called the main.
2. The caster now rolls the dice. If it's the main, they win. If it's a 2 or 3, they lose. The results for 11 or 12 are listed below.
a. If they have a main of 5 or 9, they will lose with 11 or 12.
b. If their main is 6 or 8, they lose with 11 and win with 12.
c. If they have 7 as a main, they will win with 11 and lose with 12.
3. If the caster neither loses nor wins, the number thrown is called the chance. They throw the dice again and one of these three things will happen:
a. If they roll the chance again, they win.
b. If they roll the main, they lose.
c. If they roll neither, then they keep rolling until they get one or the other.
4. The caster remains the caster until they lose three times in succession, after which they pass the dice to the player to their left, who then becomes the next caster.
Images
The pair of dice shown in the image on the far right are the only game pieces required for my game. The die on the right was made by me from casting resin. The image on the far left shows the mold I made for the die as well as the die I made. The center image is a Renaissance-era painting which shows three men playing hazard. The man in the middle is most likely the caster.
Probability Analysis
.Questions:
1. If one had a main of 7, what are the chances of winning? How different would they be compared to any other main?
2. What is the probability that the chance is rolled in a game?
3. Were the dice biased in any way? If so, which number(s) were they skewed towards, and what constitutes biased dice?
1) To answer question one, I rolled the dice 10 times with each main to see which one gives a higher chance of winning the game. As I was rolling the dice, I observed that they themselves were slightly skewed because 6 and 7 were the numbers that came up the most. This could have been due to the size and weight difference between the dice. The main that had the most wins was 7, because I found that it had 7/10 wins. This proves that it does matter which main you pick, and 7 is the best one because it has a higher chance of winning the game multiple times than any of the other mains.
2) For question two, I played 10 games of hazard and recorded in a table whether I rolled the chance or not. Out of the ten games I played, I rolled the chance in 4 of them. The reasons for not rolling a chance are rolling and getting the main, rolling and getting a 2 or 3 and losing, or rolling and getting an 11 or 12, whereupon the game is won or lost depending on the chosen main. Thus I conclude that due to the presence of so many other events which could also take place, the likelihood of rolling a chance in a single game is quite small.
3) Yes, I believe that the dice were biased, as I mentioned earlier in the answer to question one. I observed that the dice skewed towards the numbers 6 and 7, and I believe that this is so because the die that I made out of casting resin had to be sanded because one of the faces didn't turn out quite right, so that may have unbalanced the die and the other die that I used was made of a different material than my die, and that weight difference could have caused it to skew toward different numbers than my die. Biased dice are essentially dice that have been shaved on a few sides, weighted, or otherwise tampered with to result in unfair odds. These die were not intentionally loaded as there were just craftsmanship errors.
In order to obtain the aforementioned data to address the questions, I performed a series of experiments. For question one, I rolled the dice ten times each time with a different main to see whether or not there was an optimal main to use in order to gain more wins in the game. I was looking for the number of wins and losses for each main in order to find the above result. The images below show my data tables for these experiments.
1. If one had a main of 7, what are the chances of winning? How different would they be compared to any other main?
2. What is the probability that the chance is rolled in a game?
3. Were the dice biased in any way? If so, which number(s) were they skewed towards, and what constitutes biased dice?
1) To answer question one, I rolled the dice 10 times with each main to see which one gives a higher chance of winning the game. As I was rolling the dice, I observed that they themselves were slightly skewed because 6 and 7 were the numbers that came up the most. This could have been due to the size and weight difference between the dice. The main that had the most wins was 7, because I found that it had 7/10 wins. This proves that it does matter which main you pick, and 7 is the best one because it has a higher chance of winning the game multiple times than any of the other mains.
2) For question two, I played 10 games of hazard and recorded in a table whether I rolled the chance or not. Out of the ten games I played, I rolled the chance in 4 of them. The reasons for not rolling a chance are rolling and getting the main, rolling and getting a 2 or 3 and losing, or rolling and getting an 11 or 12, whereupon the game is won or lost depending on the chosen main. Thus I conclude that due to the presence of so many other events which could also take place, the likelihood of rolling a chance in a single game is quite small.
3) Yes, I believe that the dice were biased, as I mentioned earlier in the answer to question one. I observed that the dice skewed towards the numbers 6 and 7, and I believe that this is so because the die that I made out of casting resin had to be sanded because one of the faces didn't turn out quite right, so that may have unbalanced the die and the other die that I used was made of a different material than my die, and that weight difference could have caused it to skew toward different numbers than my die. Biased dice are essentially dice that have been shaved on a few sides, weighted, or otherwise tampered with to result in unfair odds. These die were not intentionally loaded as there were just craftsmanship errors.
In order to obtain the aforementioned data to address the questions, I performed a series of experiments. For question one, I rolled the dice ten times each time with a different main to see whether or not there was an optimal main to use in order to gain more wins in the game. I was looking for the number of wins and losses for each main in order to find the above result. The images below show my data tables for these experiments.
I also conducted an experiment for the probability of rolling a chance in a single game. I played 10 games in this experiment and recorded a "yes" in the left-hand column if a chance was rolled and a "no" if it was not. The data table for the chance experiment is displayed below.
For this probability analysis, I feel that the Habit of a Mathematician that would be most relevant to my approach would be Conjecture and Test, because in order to find the result that made the most sense, I had to test several different variables. I tried a different approach to address my question each time until I finally decided on recording and obtaining my data through data tables. A tree diagram representing all my data points is shown below.
Reflection
In all honesty, I felt that this project was borderline challenging for me because statistics and probability is not normally my strong suit or my most favorite subject in math. I was, however, able to overcome my initial dislike of the subject and came to enjoy learning about probability. One project I especially enjoyed doing was the Renaissance game research. I liked learning about the history behind games of chance and how they have contributed to popular games today. In general, this whole unit on probability has helped me overcome whatever difficulties I had in understanding the topic.