2503 Words11 Pages

Monty Hall Problem: a mathematical exploration of the problem
Marlou Dinkla 000768-008
Math Internal Assessment
American School of The Hague
Word Count: 2500
If you are into brainteasers you might have come across the Monty Hall problem. Another place were you might have stumbled upon it is on science programs like Mythbusters. In talking to my brother about his university math class, we came upon enigmas that involved math. Remembering back to my days in front of the television, I was reminded of a Mythbusters episode were they tried out the Monty Hall problem. Unbeknown to me at the time, was the ability to apply statics in finding out the best course of action.
After doing additional research, I learned that it originated*…show more content…*

Imagine three doors, behind one of the doors is a prize behind the other two there is nothing. You are asked to pick a door; the host opens another door with no price and asks if you want to change door and go to one that is still remaining. Now, do you stay with your original decision or do you change doors? I aim to recreate the problem with the same setup and see through a mathematical standpoint as to what a person should do when presented with a Monty Hall problem. Starting with a basic idea of how the problem works including visuals. Now, the person picks a door. For this example it will be door 1. As followed the host decides to open a door with no prize in it, in this case door 2. After opening the door the host presents the person with the following option, does he want to switch door and open door 3 or does he stay with his original pick and open door 1. Lets find out what happens with the two options he now has. The options can play out in multiple ways, shown in table one, a check mark means that is what happens and an x means that did not. To better understand figure 1 demonstrated both options. Table*…show more content…*

The probability of the Monty hall problem can be calculated with the use of Bayes Theorem (Bayes). Before I do however, let me first show how the theorem is deduced, which will also give an understanding of what it is. First it starts with the basic equation of conditional probability. P(A∩B)=P(B)∙P(A│B). The conditional probability of both event A and B happening is calculated by the probability of the event B multiplied by the probability of event A given event B happens. The next step is realizing that this equation can also be turned around. P(A∩B)=P(A)∙P(B|A). Since both the conditional probability equations are equal to P(A∩B), the equation can be rewritten as follows. P(B)∙P(A│B)= P(A)∙P(B|A) As a next step both sides are divided by P(B). This results in the following equation: P(A│B)= (P(A)∙P(B|A))/(P(B)) This is the formula of Bayes Theorem, stating the probability of event A given event B happens is equal to the probability of event A multiplied by the probability of event B given event A has happened, all divided by the probability of event

Imagine three doors, behind one of the doors is a prize behind the other two there is nothing. You are asked to pick a door; the host opens another door with no price and asks if you want to change door and go to one that is still remaining. Now, do you stay with your original decision or do you change doors? I aim to recreate the problem with the same setup and see through a mathematical standpoint as to what a person should do when presented with a Monty Hall problem. Starting with a basic idea of how the problem works including visuals. Now, the person picks a door. For this example it will be door 1. As followed the host decides to open a door with no prize in it, in this case door 2. After opening the door the host presents the person with the following option, does he want to switch door and open door 3 or does he stay with his original pick and open door 1. Lets find out what happens with the two options he now has. The options can play out in multiple ways, shown in table one, a check mark means that is what happens and an x means that did not. To better understand figure 1 demonstrated both options. Table

The probability of the Monty hall problem can be calculated with the use of Bayes Theorem (Bayes). Before I do however, let me first show how the theorem is deduced, which will also give an understanding of what it is. First it starts with the basic equation of conditional probability. P(A∩B)=P(B)∙P(A│B). The conditional probability of both event A and B happening is calculated by the probability of the event B multiplied by the probability of event A given event B happens. The next step is realizing that this equation can also be turned around. P(A∩B)=P(A)∙P(B|A). Since both the conditional probability equations are equal to P(A∩B), the equation can be rewritten as follows. P(B)∙P(A│B)= P(A)∙P(B|A) As a next step both sides are divided by P(B). This results in the following equation: P(A│B)= (P(A)∙P(B|A))/(P(B)) This is the formula of Bayes Theorem, stating the probability of event A given event B happens is equal to the probability of event A multiplied by the probability of event B given event A has happened, all divided by the probability of event

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