probability portfolio
This probability portfolio is basically a reflection on the past unit of probability in math class. For the past few weeks we have been learning about theoretical and experimental probabilities and exploring different possibilities by visiting different problems that would try and push our theoretical and experimental reasoning.
(1)
This unit involved two main approaches to most of our problems, theoretical and experimental. A problem I like that helped me understand theoretical and experimental probability better was the Game of Pig. In the Game of Pig you role an ordinary six-sided die labeled 1-6. Now every turn consists of you or the opponent rolling the die one or more times, you keep rolling till you decide to stop or if you hit 1. If you choose to stop before you roll 1 your score is the sum of all the previous numbers you rolled however when you hit 1 your score for that round is 0. This game is a great example of experimental probability because in the beginning you need to experiment to get results to solve for a theoretical probability. In the first couple of rounds for me it was all about collecting data making charts and trying to figure out what the most reasonable strategy was.
Moving to theoretical probability, in this problem I would say theoretical probability was as equal as experimental just because I thought that I needed both to make sure I was going to be correct. After experimenting and following the meaning of theoretical probability, the probability that a certain outcome will occur, as determined by reasoning, I concluded that there is a 1/6 chance of me rolling a 1 on the die so why not just take the chance. My strategy in the end was to keep rolling till I would have the highest number in the group that I was playing against and not to worry too much about that 1.
The Game of Pig definitely helped me with my other probability problems because I learned a new method for solving probability problems. My method was to always try starting with some sort of experimental solutions right off the bat and then using more data and info create a theoretical probability to either help prove or fix my experimental solution.
(2)
The piece of work that I am most proud of in this unit probably has to be the counters problem. The reason I am proud of this project is because I feel like I explored a lot and also pushed myself to find ways to use experimental and theoretical probabilities. In this game we were given a board with 11 boxes numbered 2-12. We were each given 11 circle counters to place anywhere on the board, allowed to have more than one occupying one box. As we rolled two dice we added their sums and took out a counter on the corresponding box (example: if you roll 3 and 4 then you would remove a circle counter from 7). The winner was the first one to remove all counters from the game board. We also had to come up with a strategy for winning in this game. One strategy that I had, which was later discovered by other classmates was that the numbers in the middle of the board, 6 7 and 8 were the greatest occurrences, so it was better to put counters around those numbers.
(3)
Two habits of a mathematician that I probably used throughout this unit are "Experiment Through Conjectures" and "Be Confident, Patient, and Persistent". I can't think of one specific piece of work that these habits shined like crazy but I did feel like they were reoccurring habits throughout the unit. Experimenting through conjectures was probably more dominant through the unit because of the experimental probabilities. I kept having too try out different methods before finalizing a solution, this required multiple tests and failures. This is where being confident, patient and persistent comes in. I sometimes didn't know where to start or where to continue on after a failure but I always tried to keep going and just writing stuff down like things that I knew from the problem or even just numbers that were in the problem. This somehow helped me get through the rough parts of the unit and still get something out of it.
(1)
This unit involved two main approaches to most of our problems, theoretical and experimental. A problem I like that helped me understand theoretical and experimental probability better was the Game of Pig. In the Game of Pig you role an ordinary six-sided die labeled 1-6. Now every turn consists of you or the opponent rolling the die one or more times, you keep rolling till you decide to stop or if you hit 1. If you choose to stop before you roll 1 your score is the sum of all the previous numbers you rolled however when you hit 1 your score for that round is 0. This game is a great example of experimental probability because in the beginning you need to experiment to get results to solve for a theoretical probability. In the first couple of rounds for me it was all about collecting data making charts and trying to figure out what the most reasonable strategy was.
Moving to theoretical probability, in this problem I would say theoretical probability was as equal as experimental just because I thought that I needed both to make sure I was going to be correct. After experimenting and following the meaning of theoretical probability, the probability that a certain outcome will occur, as determined by reasoning, I concluded that there is a 1/6 chance of me rolling a 1 on the die so why not just take the chance. My strategy in the end was to keep rolling till I would have the highest number in the group that I was playing against and not to worry too much about that 1.
The Game of Pig definitely helped me with my other probability problems because I learned a new method for solving probability problems. My method was to always try starting with some sort of experimental solutions right off the bat and then using more data and info create a theoretical probability to either help prove or fix my experimental solution.
(2)
The piece of work that I am most proud of in this unit probably has to be the counters problem. The reason I am proud of this project is because I feel like I explored a lot and also pushed myself to find ways to use experimental and theoretical probabilities. In this game we were given a board with 11 boxes numbered 2-12. We were each given 11 circle counters to place anywhere on the board, allowed to have more than one occupying one box. As we rolled two dice we added their sums and took out a counter on the corresponding box (example: if you roll 3 and 4 then you would remove a circle counter from 7). The winner was the first one to remove all counters from the game board. We also had to come up with a strategy for winning in this game. One strategy that I had, which was later discovered by other classmates was that the numbers in the middle of the board, 6 7 and 8 were the greatest occurrences, so it was better to put counters around those numbers.
(3)
Two habits of a mathematician that I probably used throughout this unit are "Experiment Through Conjectures" and "Be Confident, Patient, and Persistent". I can't think of one specific piece of work that these habits shined like crazy but I did feel like they were reoccurring habits throughout the unit. Experimenting through conjectures was probably more dominant through the unit because of the experimental probabilities. I kept having too try out different methods before finalizing a solution, this required multiple tests and failures. This is where being confident, patient and persistent comes in. I sometimes didn't know where to start or where to continue on after a failure but I always tried to keep going and just writing stuff down like things that I knew from the problem or even just numbers that were in the problem. This somehow helped me get through the rough parts of the unit and still get something out of it.