In math class this semester, we have been focusing all our attention towards probability and chance. Everyday, we have analyzed problems that are associated with probability, then we go in depth and discuss the problems as a class to learn more about it. Because of these in-depth explanations on the topic, I feel relatively comfortable with probability however not fully confident. We learned all sorts of different topics of probability and put them together. We started with a worksheet called "game of pig," which had to do with the probability of rolling dice. Some of the concepts we learned were:
For the renaissance project, we had to pick games that were played during the time period and we had to look into the probability of each game. The game I picked was Bingo. It originated in circa fifteen-hundred A.D. in Italy. It was not a very popular game at the time and often produced more than one winner of the game at a time since the number sheets used to be very limited in size and order. Bingo was normally played in people's homes and not commonly out in the streets like dice and card games since Bingo requires more than one board to play on. It is not known who was the first to create the game of Bingo, however it has lasted throughout the centuries. It was recreated in 1929 by Edwin S. Lowe - an American toy maker who then named it Beano, which later was altered to Bingo. The modern version only yields a single winner at a time and has a larger board with more number combinations. I picked this game because of my familiarity with it. I was comfortable selecting it because I am used to the modern version which is similar to the original from the renaissance. The game is played like so: each player gets a board with a random pattern of numbers on a 5x5 pad of numbers. Numbers are called out at random and if a player has a number that is called, they mark it off their board. The first player with 5 marked in a row, wins. The way I made it had numbers 1 through 25 in random order printed in 5x5 squares on a sheet of paper. The probabilities and permutations are near incalculable because there are an immense amount of factors that go into a game of bingo. Everything in the game is random and there are so many elements and factors where the chance of winning is always one person, but the permutations are near endless.
- Observed Probability - An observed value of an event
- Theoretical Probability - The expected and estimated probability
- Conditional Probability - The probability of an event which follows and is associated with a prior event
- Probability of Multiple Events - The probability of more than one event
- Expected Value - A predicted value sum of any number of events
- Two-Way Tables - A two-way table of counts organizes data about two categorical variables. Values of the row variable label the rows that run across the table, and values of the column variable label the columns that run down the table.
- Tree Diagram - a diagram that shows all the possible outcomes of an event.
- Joint Probability - A joint probability is a statistical measure where the likelihood of two events occurring together and at the same point in time are calculated. Joint probability is the probability of event Y occurring at the same time event X occurs.
- Marginal Probability - the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables in the subset without reference to the values of the other variables.
For the renaissance project, we had to pick games that were played during the time period and we had to look into the probability of each game. The game I picked was Bingo. It originated in circa fifteen-hundred A.D. in Italy. It was not a very popular game at the time and often produced more than one winner of the game at a time since the number sheets used to be very limited in size and order. Bingo was normally played in people's homes and not commonly out in the streets like dice and card games since Bingo requires more than one board to play on. It is not known who was the first to create the game of Bingo, however it has lasted throughout the centuries. It was recreated in 1929 by Edwin S. Lowe - an American toy maker who then named it Beano, which later was altered to Bingo. The modern version only yields a single winner at a time and has a larger board with more number combinations. I picked this game because of my familiarity with it. I was comfortable selecting it because I am used to the modern version which is similar to the original from the renaissance. The game is played like so: each player gets a board with a random pattern of numbers on a 5x5 pad of numbers. Numbers are called out at random and if a player has a number that is called, they mark it off their board. The first player with 5 marked in a row, wins. The way I made it had numbers 1 through 25 in random order printed in 5x5 squares on a sheet of paper. The probabilities and permutations are near incalculable because there are an immense amount of factors that go into a game of bingo. Everything in the game is random and there are so many elements and factors where the chance of winning is always one person, but the permutations are near endless.
Since there are thousands plus factors that go into winning a game of bingo, one simple cannot draw a tree diagram. Pictured below would be the start of a game with a single player. A bingo board has 25 numbers to start with on the sheet which means a number pulled could be 1 number out of 25. This simple start leads into many other branches with many other outcomes. Calculating the real probability is understandably a task too great to undertake. The habits of a mathematician that I utilized to complete this endeavor, was being systematic. If I hadn't focused on being systematic, I'm not sure I would have been able to manage so many different variables all at once. If I had a specific question, for the sake of probability in bingo, here's an example: What is the probability of getting number 4 on the second round of bingo? An easy way to solve this problem would be to draw out a tree diagram. Shown below, the probability of not rolling a 4 on the first round (in order to get it on the second round) would be 24/25. After that, the probability of rolling 4 on the second round (our desired outcome,) would be 1/25 which is what we want. The overall total probability would be all of the probabilities multiplied together. 24/25 x 1/24 = 1/25. Done. A habit of a mathematician that could be employed to solve a problem like this would be to be systematic. The systems and strategies we use would be the probability tree diagram and the method of multiplying the outcomes in order to get what we want. These are both two systems that we could use to solve this. Starting one step at a time and picking one of two systems to mentally compartmentalize it.
Throughout this project, I really do believe that I have learned a lot. I now have a more in-depth understanding of probability and what goes into it. I learned a load of information about the renaissance - before, I did not even know that games like the ones we commonly play today were invented way back in 1300-1600. I feel like I've grown as a student with the independent critical thinking I've had to do with all of the work sheets we were given. I did initially have a great challenge with understanding writing out the format of probability equations and the function of probability trees, but after it all clicked, simultaneously it began to make more and more sense, especially since we went so in-depth with each problem. This was definitely a great challenge in the project. At first everything was near incomprehensible. I have noticed that only in math or physics classes (basically everything besides social studies classes), I have had a struggle to understand until one day it all comes together at once to make more sense than it did before. A success I had with this project was getting everything turned in on time and retaining a quality standard. Overall, this project had its ups and downs, yet still worked out in the end as a good tool for education and personal growth.
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