Playing with History
Probability: the extent to which something is probable; the likelihood of something happening or being the case.
We started the unit of probability with the Game of Pig. With little to no knowledge of probability, we started with observed probability which is when you’re interested in a certain outcome and conduct the test yourself. For example, one of the first worksheets we did was called “Who is Cheating?” and you had to roll a die to figure out if the subject, or athlete in this case, was doing a certain thing, doping. Once you figured out whether or not they were doping, you then had to either roll another die or flip a coin three times. This gave us the data we then put into a tree diagram, which is a way to keep your work organized and makes it easier to understand, and the proportions and probabilities of a certain outcome. Next, we’d put it into a two way table which is another way we learned how to set up our data. This led us to learning about marginal probability which is when you add up the two columns or rows in a two way table to make Pr[A] or Pr[B] and joint probability which is when you add the two columns together, Pr[A and B]. Once we learned how to calculate observed probability and marginal probability, we moved on to calculating predicted probability or theoretical probability which is when, instead of running the experiment yourself, you use math to calculate what should happen based off probability and proportions. The probability of multiple events is when you are interested in more than one outcome that have no correlation or effect on each other. For example, if you were to flip a coin three times, each time you have a 50/50 chance of getting a heads or a tails each time. On the other hand, however, conditional probability is when you’re interested in an outcome where one event directly affects the other. For example, picking marbles out of a bag without replacing each marble, each time the chances of picking a certain marble changes. To wrap up the unit we ended with “The Breast Cancer Problem” which allowed us to use all of the skills we learned and figure out what the conditional probability of a woman receiving a false-positive on her breast cancer test. Finding expected value is when you find the probability of an event happening (event B) after another event has already happened (event A) so in this particular problem we were solving for the probability that a woman tested positive given that fact that she was negative.
Below are some examples of my work and calculations!
Probability: the extent to which something is probable; the likelihood of something happening or being the case.
We started the unit of probability with the Game of Pig. With little to no knowledge of probability, we started with observed probability which is when you’re interested in a certain outcome and conduct the test yourself. For example, one of the first worksheets we did was called “Who is Cheating?” and you had to roll a die to figure out if the subject, or athlete in this case, was doing a certain thing, doping. Once you figured out whether or not they were doping, you then had to either roll another die or flip a coin three times. This gave us the data we then put into a tree diagram, which is a way to keep your work organized and makes it easier to understand, and the proportions and probabilities of a certain outcome. Next, we’d put it into a two way table which is another way we learned how to set up our data. This led us to learning about marginal probability which is when you add up the two columns or rows in a two way table to make Pr[A] or Pr[B] and joint probability which is when you add the two columns together, Pr[A and B]. Once we learned how to calculate observed probability and marginal probability, we moved on to calculating predicted probability or theoretical probability which is when, instead of running the experiment yourself, you use math to calculate what should happen based off probability and proportions. The probability of multiple events is when you are interested in more than one outcome that have no correlation or effect on each other. For example, if you were to flip a coin three times, each time you have a 50/50 chance of getting a heads or a tails each time. On the other hand, however, conditional probability is when you’re interested in an outcome where one event directly affects the other. For example, picking marbles out of a bag without replacing each marble, each time the chances of picking a certain marble changes. To wrap up the unit we ended with “The Breast Cancer Problem” which allowed us to use all of the skills we learned and figure out what the conditional probability of a woman receiving a false-positive on her breast cancer test. Finding expected value is when you find the probability of an event happening (event B) after another event has already happened (event A) so in this particular problem we were solving for the probability that a woman tested positive given that fact that she was negative.
Below are some examples of my work and calculations!
The Renaissance Project Probability Game
Alouette
My Renaissance game was called Alouette. It was first documented by Francois Rabelais in 16th century France. Although some believe it has a Spanish origin, the French deck used was created long before the Spanish deck. Being a game played by everyone from royalty to the lower classes, it was played in a lot of homes so the traditional Alouette deck could be found in most residential areas. A modern version of this game that lots of people know today is War. The two games are very similar because they both consist of flipping cards to see who can outnumber the other player. The main difference in the two games is the different decks. War is played with a standard 52-card deck while Alouette is played with a special "Alouette deck" that doesn't have any 10's and has a different scoring system. It requires 2 to 4 players and was recommended to be played in two teams of 2 people. I decided to choose this specific card game to show for exhibition because of the many ways to analyze its probability.
Because the game is played like war, it's fairly straightforward on how to play. The players take turns flipping over cards from their deck of nine Alouette cards. The player with the highest card value of all the players takes those cards and then you continue with the game. There were three ways people would decide who won, whoever had the most cards at the end, whoever had the 3 of diamonds, or whoever had the Jack of spades would automatically lose. For exhibition I adapted the game by simply using a standard 52-card deck but I still applied the same rules. I did this so it'd be easier to calculate or analyze the probability of certain things and so it's be easier to explain to people at exhibition.
Chance or probability plays a huge role in my game. Each turn a player has a certain probability of winning or losing that round. With each round, the chances of winning or losing change because the cards aren't replaced until the game is over.
Alouette
My Renaissance game was called Alouette. It was first documented by Francois Rabelais in 16th century France. Although some believe it has a Spanish origin, the French deck used was created long before the Spanish deck. Being a game played by everyone from royalty to the lower classes, it was played in a lot of homes so the traditional Alouette deck could be found in most residential areas. A modern version of this game that lots of people know today is War. The two games are very similar because they both consist of flipping cards to see who can outnumber the other player. The main difference in the two games is the different decks. War is played with a standard 52-card deck while Alouette is played with a special "Alouette deck" that doesn't have any 10's and has a different scoring system. It requires 2 to 4 players and was recommended to be played in two teams of 2 people. I decided to choose this specific card game to show for exhibition because of the many ways to analyze its probability.
Because the game is played like war, it's fairly straightforward on how to play. The players take turns flipping over cards from their deck of nine Alouette cards. The player with the highest card value of all the players takes those cards and then you continue with the game. There were three ways people would decide who won, whoever had the most cards at the end, whoever had the 3 of diamonds, or whoever had the Jack of spades would automatically lose. For exhibition I adapted the game by simply using a standard 52-card deck but I still applied the same rules. I did this so it'd be easier to calculate or analyze the probability of certain things and so it's be easier to explain to people at exhibition.
Chance or probability plays a huge role in my game. Each turn a player has a certain probability of winning or losing that round. With each round, the chances of winning or losing change because the cards aren't replaced until the game is over.
Probability Analysis
What's the probability of winning the round if a person flips a certain card?
Here I chose to analyze the probability of a person winning after the first person flipped over a seven. In this case it is important to remember that anything lower than a seven will result in a loss and that there are three cards that are also sevens and would depend on the next round to determine who takes the cards. With that in mind you just count up the remaining cards that you can win with and you get the probability of winning after the first person flips a seven.
I decided to use the Habit of a Mathematician: Start Small because there were so many ways to go about analyzing this.
Because I thought this was a bit on the easier side, I decided to try again using a different Habit and a different question. The question I tried solving was "What's the probability of someone winning with the three of diamonds?"
What's the probability of winning the round if a person flips a certain card?
Here I chose to analyze the probability of a person winning after the first person flipped over a seven. In this case it is important to remember that anything lower than a seven will result in a loss and that there are three cards that are also sevens and would depend on the next round to determine who takes the cards. With that in mind you just count up the remaining cards that you can win with and you get the probability of winning after the first person flips a seven.
I decided to use the Habit of a Mathematician: Start Small because there were so many ways to go about analyzing this.
Because I thought this was a bit on the easier side, I decided to try again using a different Habit and a different question. The question I tried solving was "What's the probability of someone winning with the three of diamonds?"
Because 52 is easily divisible by 4, I switched the amount of players from 2 to 4 in hopes of making the numbers a little easier to deal with. I wasn't sure how to go about solving this problem so I decided to use the Habit of a Mathematician: Conjecture and Test. First I started with a probability tree, I figured out the probability for winning the first round if each of the cards were different but not the probability for winning the first round, as there are cards that are the same, or for the specific card in my proposed question so I decided to try something else. Using another Habit of a Mathematician, Be Confident, Patient, and Persistent, I began to solve it another way. I started by figuring out how many combinations of person A having the Ace card there were in total and ended up finding six. I then multiplied each column by six to show each player having that combination for themselves. After that, I multiplied that large number (6*6=36*6=212*6=1,296) by 4 to get the total number of combinations for 1-4 and then multiplied that by 3 to get the combinations for 5,6,7,8,9,10,11, and 12 which equaled 15,552. Then because there's actually 13 combinations, I added an extra 1,296 and got 16,848. This led me to believe that one player had a 1 in 16,848 chance of keeping the 3 of diamonds, however, I realized that that didn't really make much sense as obviously there's a winner every time and it's not very difficult to win. I did try to solve the problem though and I will continue to look into solving it.
Reflection
All in all, I think I did really well on this project. In the math part specifically, I did get a little confused on some parts because the terms were so similar but when it came down to actually solving things and figuring out the probabilities of certain outcomes, I really did good. I feel like the project was a little long for my learning style which made it a bit challenging at times but I have learned a lot from this project.
All in all, I think I did really well on this project. In the math part specifically, I did get a little confused on some parts because the terms were so similar but when it came down to actually solving things and figuring out the probabilities of certain outcomes, I really did good. I feel like the project was a little long for my learning style which made it a bit challenging at times but I have learned a lot from this project.