# Probability and Monty Hall Poblem

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Probability and risk management are involved in everything that we do.  They make up the science of chance.  I am currently taking a course here at Penn State on Probability Theory and it has really sparked my interest.  The basic concept of probability lies in the likelihood of an occurrence of an event.  In class, for example, events can be categorized as event A and event A'.  This is an example of a binomial distribution of events.  Since there is no third outcome, the probability of event A occurring is equal to the probability of event A' not occurring.

Many times we are torn between the choice to do something or to not do something.  This would be considered a real-life binomial distribution of chance.  It is the most common type of distribution in our day-to-day lives and one of the more simple to understand.  The risk management part comes in when there is a reward involved with the occurrence.  In a sense, the analysis of risk is weighing your options.  In our lives the probability of an event corresponds to the risks and rewards we place on a decision.

Perhaps this decision is whether to go out or stay in and catch up on work.  Lets say that event A is going out and event A' is to not.  On a Monday, there is very little reward for event A being that there won't be as many people going out as there would be on a weekend night.  During the weekend, there is a higher reward for event A since you will have more fun being around more of your friends than during the week.  These are the rewards of going out relative to the day of the week.  The risks associated with these days are inversely related to the rewards.  To go out means to lose the opportunity to study and we determine the probability of event A occurring based on the reward of choice A.

In class we learned that due to the Law of Large Numbers probability would determine the choice we should make in the future.  In the long run, the choice will waver towards whatever event has the highest probability.

In Science 200, we spoke about the game show problem from Let's Make A Deal.  The decision at first was to pick one of three doors in hopes of getting the reward (event A) of a new car behind one door and not a goat (event A') behind two of the doors.  The probability at first is simply one (reward) out of three choices.  This is a 33% and there was very little disputing it.

After a choice was made, one of the other two doors was opened to reveal a goat.  Following the unveiling of a goat, contestants were offered the chance to switch doors.  This choice at first appeared to be a 50/50 shot at a new car and many quickly dismissed statistics and probability.  Instead they thought with their hearts rather than brains.  The stubbornness of many contestants to stick to their guns and keep the first door resulted in often times, a goat.

Over time a trend began to occur in which the decision to stay resulted in a 33% chance of reward and not a 50% chance that was assumed.  The Law of Large Numbers dismisses bad luck in this situation and statisticians looked for a true reason behind the probability.  It was first proposed by Steve Selvin in a letter to the American Statistician that the correct choice was to switch doors.  This was because the game show host knew which door the car is behind and he knew to pick the door with the goat in it to display to you.  At first the probability of your door having a car was 33% (event A) and a 67% chance that the car was behind the other two doors (event A').  Since the host eliminated half of the doors from the probability of A' the full probability landed on the door which he did not display.  The door that you chose at first maintained its original probability while the other door doubled its chances on winning.  This image captures the probability of winning given that you switch doors:

Looking at it now it seems like a no brainer to switch doors and take the two thirds odds.  This concept is not easily grasped and led to debate over the topic but in the end, you can't argue with the science involved.  Probability is all around us and is something that should be taken advantage of with our every day decision-making.

BBC Article Explanation

Demonstration of the Monty Hall Problem

Statistical Proof of the Monty Hall Problem

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When I first heard about this, I was so confused and my immediate answer to the problem was: stick with the door with you have. It's done pretty well so far, so why not just stick with it to the end? Looking at this now it seems to easy to win and yet , ignoring the 1/3 chance of failure, there is one other major factor that stands in our way of winning: Human nature.

Like Andrew said in class humans are averse to taking responsibility for their actions when they have negative consequences so it's possible that someone might go in to the show and even though they know of this may choose not to switch doors, simply because in case they choose wrongly, they won't be to blame, strange isn't it?

On a side note, I read this interesting paper that shows the possibility of two alternate variations of the Monty Hall problem. I especially liked the Monty Fall problem!

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