Overview: During the course of this unit, all of our assignments consisted of playing with die, coins, and anything else to do with chance as well as finding probability of different factors. To explain all of what we were expected to learn in a little bit more detail, I will be using one of the worksheets we were assigned as an example. The worksheet is called "Who's Cheating?" Which looks like this:
In summary, we were given a problem with 36 athletes who have to take a doping test and find the probability that an athlete will be accused or get away with cheating. The first thing that we needed to know is that probability is the likelihood of something happening so, the question, in other words would be: What is the likelihood that an athlete will be accused or get away with cheating?
On the assignment there is a tree diagram that represents all of the possibilities for each athlete according to what number is rolled on the die or flipped on a coin. The tree diagrams shows that the athlete is either doping or not doping, then they can either get accused or not accused. When it comes to visualizing possibilities as well as probabilities, the tree diagram is a very helpful tool. Making a tree diagram can help you a lot when it comes to expected value. If you look back to the assignment shown above, you'll see the tree diagram that lists out all of the possible outcomes. Expected value is the outcomes multiplied by their value of probability. Pretend you've selected 5 athletes from the 36 and assigned them numbers 1-5. For all of them, there is a 1/2 probability that they are not doping. Take all outcomes (i.e. athletes 1-5) and multiply them by 1/2. For example, 1(1/2) + 2(1/2) + 3(1/2) + 4(1/2) +5(1/2).
There were two parts to "Who's Cheating?". The first part was called "Who's Cheating? - Theoretical Probability". Theoretical probability is the amount of favorable outcomes divided by (also known as "over") the amount of possible outcomes which is written like this: # of favorable outcomes/# of possible outcomes. Let's focus on the athletes that weren't doping and were accused. Let's say 5 of the 24 people who weren't doping are unfairly accused. The theoretical probability of an athlete being unfairly accused would be: 5/24. 5 is the favorable outcome because that's the amount of people falsely accused, which is exactly what we wanted to know. 24 is the greatest number of possible outcomes or people who could've been accused even though they weren't doping.
The second part was called "Who's Cheating? - Observed Probability". Observed probability is slightly different than theoretical because observed probability is what actually happened while theoretical is based on reasoning. To find the conditional probability of something, you'd have to complete the experiment before finding the actual probability that an athlete will be accused or get away with cheating. One thing that was very useful when it came to doing the experiment and understanding what future problems were asking was conditional probability. Conditional probability looks like this: Pr[A|B]. That line in the middle is thought as a symbol meaning "given that." For example: What is the probability that an athlete was accused given that they weren't doping. Which would be written like this: Pr[Accused|Not Doping].
I found that conditional and joint probability look very similar but while conditional probability looks like this: Pr[A|B]. Joint probability looks like this: Pr[A and B] which means what is the probability those 2 things are happening at the same time. For example: what is the probability that they're an athlete and they're doping? Which would be written like this: Pr[Athlete and Doping]. Joint Probability and the Probability of multiple events happening are very similar as well. Let's say that someone's an athlete and they're doping. Let's say that the probability of them being an athlete on a certain team is 1/36 and the probability that they're doping is 1/5. To find the probability using multiple events, you'd have to multiply 1/36 and 1/5 which would give you: 41/180 probability that someone was on a team with 36 athletes and was doping.
Another way to map out all of this probability is to use a two-way table which will look like this:
Above, I made a two-way table using the "Who's Cheating" problem as an example. At the bottom of the two-way table, there are boxes lining the outside that are labeled "total". All of the boxes at the bottom that have a "+" in them are the totals of the events on the inside. Those are called margins, otherwise known as "marginal probability". So if you add the 2 boxes in the middle of the lowest row, they will come to a total of 36. If you add the 2 boxes in the middle of the last column, they will equal 36 as well.
Renaissance Game: The renaissance game that I chose is called "Nine Men's Morris" also known as "Nine Man's Morris". Nine Men's Morris dates back to around 2,000 years ago close to the time of the establishment of The Roman Empire. The actual origin of the game is unknown but it was very popular during the renaissance. The game has been played all over the world. The pattern of the board has been seen in scriptures in India, in a temple in Egypt, Athens, Ireland, carved in stone by Pickering Church, carved on top of a barrel on the 16 Century Mary Rose, a board was even carved into the viking ship Gokstad. The board game was mentioned by Shakespeare and John Gower (an english poet). The game was played outside, inside, in boats, even in wood or stone. It was available to anyone as long as they could carve it into a solid surface.
There are no modern versions, the game that's played today is the same as the one played so many years ago. There are just different variations of the same game that require less game pieces.
I chose this game because it sounded very interesting. It reminded me of the game chess because of all the strategy that the game naturally requires. The game play is very tedious and can take anywhere from 2 minutes to an hour to play. I also liked the design of it and thought it'd be really cool to re-create. The rules of the game tip-toe on the line of complicated but they're not impossible to understand. Here's how to play:
Originally, there was no chance/probability in the game because it's (usually) a strictly strategy game. While making our version of the game, we decided that the die would decide how many spaces that someone would move their peg and if a 1 or a 5 was rolled, you'd lose your turn.
Images:xnmj
(Nine Men's Morris carving found in Norwich Cathedral.)
(Me and my partners' version of Nine Men's Morris.)
Probability Analysis: The question me and my partner decided to answer was: Pr[B rolls higher than A | A rolls lower than B] or what is the probability that Player B will roll higher than Player A?
Above is the probability tree of all the possibilities that Player A can roll and their outcomes, along with Player B's roll. The boxes in red represent player B's possible rolls according to what person A previously rolled. However, we used a different method to solve our problem. What we used is called "expected probability" which is very similar to expected value.
How we solved this problem was a little complicated. We set up the problem in two columns. The left column represents the sides on a die and the right represents the probability of person B rolling higher than person A. The probability of rolling 1-6 on this (fair) die is 1/6. The right column is a bit more complicated than the left. The right column's probability is the number that is rolled subtracted by 6. For example: 1/6 - 6/6 = 5/6, so on and so forth except for the probability of rolling higher than a 6 because it's not possible. Once we found all of the probabilities for the left and right columns, we had to multiply them together and then add up the multiplied probabilities. For example, 1/6(5/6) + 1/6(4/6) + 1/6(3/6) + 1/6(2/6) + 1/6(1/6) = 15/36.
Habits of a Mathematician: The habits of a mathematician that I used was: be confident, patient, and persistent, stay organized, and collaborate and listen. I had to be patient and persistent when I was learning how to figure out the probability that player B rolls higher than player A. I had to collaborate with my partner while making the game and the probability. I learned how to take a step back and that it's possible to over-complicate math problems.
Reflection: Overall, I think that this project was very useful. I feel like I am comfortable with probability, though I might struggle in a few areas. I know I've come a long way and conquered most of my confusion with conditional vs. joint probability as well as expected value. I know way more than I did about probability right now, then I did going into this project. I liked being able to research about my game and I got to realize how much of an impact that history has on the present. I liked being able to challenge myself in this DP update as well by using all the terminology we learned in the context of a single assignment which definitely wasn't easy. I challenged myself by picking a strictly strategy game and adapting probability to it then creating a problem to solve from a game that my partner and I created. In conclusion, I enjoyed many aspects of this project and I liked how much research, work, and dedication went into making sure that I got the very most out of this project as a whole.