Math IA Game of Theory: War Casualties
I have gained valuable skills and knowledge after learning topics on statistics and probability in the syllabus of IB Mathematics Standard Level. However, it has been my desires to come up with an area whereby I can apply the concepts I have learnt in the class to something that was physical and relevant in the world to promote my understanding on the topics. Today, we live in a world where we normally take risks and engage ourselves in many choices where we have to make an important decision. This situation made me think of a current challenge that has emerged between America and North Korea. There has been conflicts and tension that has been developed because of a dangerous test of missiles by North Korea that is capable of destroying many from a far distance.
After revising the topics again, I realized that there must be various actions that both countries can plan to take. As a result, I had to figure out the consequences that would occur when America attack North Korea but North Korea is not attacking America, North Korea attacks America but America is not attacking North Korea and when both countries attack each other. In these situations, we can apply probability to determine the result of each outcome.
Aim of exploration
My main goal in the investigation is applied a probability game that will use Geogebra outside the lesson environment. The analysis will assist in solving an equation that will aid in producing a value that will have to mean in the activities of the world. It will create will solve and provide an answer to the question that is happening today, instead of solving the problems in the syllabus.
Overview of game theory and Probability
Game theory enables us to understand models of mathematics that will involve cooperation and conflict between intelligent rational decision-makers. In this experiment, we will apply real-world challenges of warfare between America and North Korea. On the other hand, the probability is one of the branches of mathematics where the possibility of an event to occur can be evaluated. Numerical measures ranging from 0 and 1 are used in the field of probability. Mostly, 0 is usually used to indicate the impossibility while 1 can be used to highlight the certainty of an event to happen.
In the field of probability, there are various types that describe different events and are applicable. Therefore, my understanding of the concept of probability will help me to analyze and implement the data more efficiently. To begin with, we can represent the situation our case whereby we have two events that cannot occur at the same time. It is either America attacking North Korea but North Korea does not attack America or North Korea attacks America but America does not attack North Korea. We can represent it as disjoint events whereby two events cannot occur at the same time.
Let assume that event A represent America and event B represent North Korea. Therefore;
Event A=getting consequence X
Event B=getting consequence Y
The sample space for both events is X and Y. It is notable that in both cases event A and B cannot occur together. These two conditions can be represented using Venn diagram.
Conditional probability can be used when the probability of some event A, given the occurrence of the other event B. in this situation, we try to calculate the chances of the second event to happen when given that the first event has already occurred. In this case, we can highlight that the probability of an event A is dependent on the occurrence of event B. Therefore, we can represent the probability as P (A/B) such that,
P (A/B) = P (A^B)/P (B)
P (A^B), represents the probability of event B, P (B) and event A, P (A). This indicates clearly that both are independent. However, we can arrange the same equation in another way to get the following equation. P (A^B) = P (A/B) x P (B). However, we can interchange the events A and B to get our new probability as P (B^A) =P (B/A) x P (A).
It is good to note that Bayes'probability rule indicates that P (A^B) = P (B^A). After establishing the equation, we can apply Monty Hall problem. The problem indicates that when you are on a game show and you are given a chance to choose from three doors and one door has a goat behind it. In this situation, the choice you make in opening the doors is what will earn you a goat. Different outcomes of various possible combinations of choices and the decision of the player will be shown in a table. In our case, we can have two players who are America and North Korea. Their choices of getting involved in the fight will have effects towards their rivals. For instance, the choice that each party makes can determine whether they will lose or win the battle.
The second column indicates the exact location where the country selects and gets the ammunition. In the third column, it contains the choice of the player whether select one, two or three. The fourth column contains the final choice of Monty function which is set to open a particular door. The data has to be set with a function that will select randomly pick the door that hides the weapons. In addition, there will be calculation of probabilities of the winning players when they change their decisions to pick a new door. In this situation, we can see that when a country chooses the first door as their first choice, Monty can choose to open either of the doors. If the player sticks to the decision, one is likely to change, and then they might lose. However, when they stick, they will definitely win.
From the three doors, the probability of sticking and losing is P (1/3) while the chances of changing the mind and wins ifs P (2/3). However, we can prove the problem involving Monty Hall. To begin with, we can the probability of the weapons to occur in the three doors will be:
P (1) = 1/3
P (2) = 1/3
P (3) = 1/3
When the player chooses the first door, the probability that the third door will be opened on the location of the prize can be found. P (3/1) = 1/2 is the probability of Mont opening the third door when the weapons in place in the first door. But since the player has chosen the first door, Mont can decide to open either second or third door. Therefore, P (3/2) = 1 and P (3/3) = 0
From the above experiment, we can examine that when an America attacks North Korea and Korea does not attack back, the chance of casualties to occur is more on the side of Korea since they do not have access to the weapons and they are not ready to find. Therefore, the probability of Korea winning is 0 while the probability of America winning is 1. On the other hand, North Korea can attack America and America is not fighting back. Here, more casualties, destructions, and injuries will be experienced towards the American. American chances of winning the battle are 0 while Korea 100 percent of victory. Also, when we have two countries fight against each other, their probability of winning will depend on various factors. For instance, the choice will determine the chances of winning. If America makes a wrong decision of strategy and they are not willing to change they will end up losing. Also, unwillingness to shift from poor to strong strategy method among the Koreans will hinder their success.
I think the discovery of these strategies of conflicts and war will give a solution to the country that is involved in the war. Each country will understand that the country is unwilling to participate in a war, there is need to look for a better solution to end the hatred and conflicts. It is urgent for the formation of members who will try to unite various countries to promote peace, love, and unity. The stable state will ensure growth and development of trade and other businesses. Battles will increase sufferings to people. Many people will die; lose their properties and destruction of their environment.
However, I found the experiment to be fascinating not only because of understanding the strategies of winning by both switching and sticking to initial decision but to examine how we can solve problems that affect the country. We need to make the decision that will have positive impacts on people. Today, North Korea has been carrying out missiles test that has created tension and chaos. This act has led many people to intervene to stop their actions. However, if they switch their ideas to more innovative that will improve the living standards of the citizens in the country.
In doing this Internal Assessment, I was in a position to experience how the concepts of mathematics can be used in real-life situations around the world. Professional Mathematicians have applied the skills to social problems facing our societies today. Among them are threat and conflicts when two countries are involved hence creating tension among the citizens. The knowledge we can help and come up new ideas that have high chances of uniting conflicting parties.
Rosenhouse, Jason, and Jason Rosenhouse.? The Monty Hall Problem: The Remarkable Story of Math's Most Contentious Brainteaser. Oxford: Oxford University Press, 2009.
Spiegel, Murray R, John J. Schiller, and R A. Srinivasan. Probability and Statistics. New York: Schaum, 2013.
North Korea-US tensions: How worried should you be? 2017. http://www.bbc.com/news/world-asia-40882877