Favorite Puzzle (Final Draft)

Andrew Meier

10A

Favorite Puzzle (Final Draft)

4/2/13[M1]

The Monty Hall Puzzle

A Mathematical puzzles are very[M2] interesting. [M3]Theymathematical puzzle presents a problem that appears to be very difficult or maybe sometimes a problem that is difficult. However, looking at the problem from another angle often makes the solution clearonce the solution is known, it makes sense and the problem does not look as difficult as it did before[M4]. In the case of this free-throw percentage problem, it is very the puzzle appears difficult to figure out the answer at first, but once the answer is known, it makes sense.

A basketball player keeps track of her free-throw shooting percentage. A few games ago, the percentage was at some value below eighty percent. Now her average is above eighty percent. Is it possible that her average was never exactly eighty percent?

The answer for this puzzle is that her percentage had to be exactly eighty percent at one time. If you look at the puzzle in terms of the number of shots made to the total number of shots taken it will be challenging to come up with why this is the case. However, if you look at the puzzle in terms of the shots made vs. the shots missed, the answer comes into light. This is because eighty percent is a ratio of 4[M5] four shots made made shots to every one1 missed. Each shot is a whole number, so thate ratio must be reached. For example, at one point her ratio could have been 2:1. Then when she made one more it would improve to 3:1. Making one more would make it 4:1 or eighty percent. Since each shot is always worth one, there is no way to skip over this ratio. The puzzle may have appeared difficult at first, but the solution is very[M6] clear[M7]once given the solution, it is clear.

This next puzzle is my favorite because it is not like so many other puzzles. Even after hearing the solution, it does not make sense. ItIt The puzzle is known as The Monty Hall Puzzle, and I came across it while watching the movie 21.

Joseph has just made it to the final round of a game show. In front of him are three doors. Behind one of the doors is a new car, and each of the other two doors is hiding a goat. The game show host explains that the game is quite simple. Joseph gets to pick a door and take home whatever is behind it.Joseph thinks for a few minutes and picks a door. The host, knowing what is behind each of the doors, goes and opens another one of the doors revealing a goat. He then asks Joseph if he would like to change his answer, or keep the door he had originally chosen. Is it in his best interest to switch doors? And why?

The movie’s short answer [M8]to the puzzle was that originally there was a thirty three percent chance of selecting the right door. Then, after one of the doors was opened, switching doors gave the player a sixty seven percent chance of selecting the correct door. This puzzle has become my favorite because Tthis solution really[M9] stood out to me. At first glance, most people would argue that it is the same probability if you switch doors, so it doesn’t matter. Or maybe it’s even worse to switch doors because the game show host might be playing a trick on you. If that was the solution you were thinking, you wouldn’t be different than many other people, including me. This problemwas put in Parade, a magazine, and thousands of readers wrote in complaining about the solution. Of the letters received, 1,000 where signed by PhDs who claimed the solution was wrong.[1]This is what makes this puzzle so unique. The solution has been debated from the beginning, and people still do not believe it.

This puzzle has really stood out to me because Cconceptually, this puzzle, it is very hardtricky to understand, or even believe. Thousands of people thought the solution was wrong when it was published[M10] and I’m sure many people were skeptical to the answer in the movie as wellwhen they watched the movie. How can it be that changing your answer part way through can increase your chances? If you look at both all three doors [M11]before you make your decisionpick either, there is a thirty three percent chance that the car can be behind either door. So how does opening one of the other doors change this?

Mathematically, the solution can be stated very easily. When the contestant first selects a door, hehas a 1/3 chance of selecting the correct door. Then, the host opens one of the other two doors that contained a goat. Those two doors together had a combined chance of 2/3 of having the car. The contestant now knows that the car can only be in one of those two other doors, so the probability that the car is behind the other remaining door in the pair is now 2/3 instead of 1/3, making it in his favor to switch.

The solution can be better understood if we consider the three possibilities. For each of these, we will assume that the contestant selects door number 1 first and then switches when given the opportunity. Possibility A is that the car is behind door number 1. This is the unfortunate possibility where the contestant ends up changing to the goat and he loses. Possibility B is that the car is behind door number 2. The host opens door number 3 and the contestant switches to door 2 and wins the car. Possibility C is that the car is behind door number 3. The host opens door number 2 and he switches to door number 3 and wins the car. Out of the three scenarios, the contestant expects to win wins 2/3 of the times he switches expected instead of the expected 1/3.

But even after understanding the probability behind the puzzle, what would I do in the situation if I were on the game show?E After all, how do you know that the host is not trying to get you to change your answer because you chose the correct door the first time? ven after understanding the math behind the puzzle, would I really change my answer? [M12]This[M13] is what I believe throws most people off. In the puzzle, the host opens will open another door regardless of which door the contestant chooses at the beginning. If I was actually on the game show, I would have no way of knowing if the host was going to open another door every time, or just when I selected the correct door on my first guess. If the host does not want me to win the car, he wouldn’t give me another chance if I picked a goat on my first guess. [M14]

Andrew,

I can see why you found this puzzle interesting. The solution is counterintuitive, but you do an excellent job explaining why it is nevertheless correct. I think there are two related areas in which you could improve your writing. Firstly, your style was a bit dry at times. Try to use more expressive terminology, vary your sentence structure, and speak in less abstract terms. Secondly, be more conscious of your usage of intensifiers (really, very, etc.). Simply choosing better adjectives will be a big step towards addressing the first area.

-Matt

[1]Tierney, John(July 21, 1991),"Behind Monty Hall's Doors: Puzzle, Debate and Answer?",The New York Times, retrieved 2008-01-18

[M1]This would go on left for MLA (ask prof. what style he wants)

[M2]Use intensifiers sparingly. You could almost always have picked a more expressive adjective instead.

[M3]Not the most compelling hook. See if you can grab the reader’s attention, or at least immediately launch into your puzzle.

[M4]This does not inherently make mathematical puzzles interesting. Perhaps stress how the puzzle is completely reconceptualized rather than how it becomes easier.

[M5]I would spell this out since you did the same for eighty.

[M6]See above

[M7]Explain the solution a little bit more before saying it is very clear. Make sure the reader, who may not be as familiar with the puzzle, is on the same page.

[M8]Try to fit the gist of it into one sentence. Otherwise use a colon (eg. The movie’s short answer is as follows:…)

[M9]intensifier

[M10]you’ve said all of this before.

[M11]I think you mean all three

[M12]Need a better transition. This question does not immediately follow from what you just stated. In fact, it seems obvious that you would change your answer.

[M13]What does “this” refer to?

[M14]This is a valid point, but you’ve effectively introduced a new element into the puzzle.