Three Doors Problem-Monty Hall Problem in the Movie 21

“Suppose you are in a game show, and you are given a chance to choose three different door. Now behind one of the doors is a new car, and behind the other two are goats. Which door would you choose?”In the movie 21, Professor Mickey asked a question in his class. The play of the talk between Mickey and Ben impressed a lot of people. In fact, the question in the movie is a classic question – Monty Hall problem! Let’s take a look!


What the Monty Hall problem says? 

Let us continue with the question Mickey asked in the class. Ben then replied,

“Hmm… Door number one.”

“Door number one! Ben chooses door number one. The game show host , by the way, knows what’s behind the door, decides to open another door. Let’s say he choose door number three behind which sits a goat. Now Ben, would you want to stay with door number one or go with door number two? Is it in your interest to switch your choice?”

“yeah”

“Remember the host knows where the car is. How do you know he’s not playing a trick on you trying to use reverse psychology to get you to pick a goat?”

“I wouldn’t really care. I mean my answers based on statistics  based on variable.”

“We just asked you a simple question.”

“Yeah, but it changed everything. When I was originally asked to choose a door, I have 33.3% chance of choosing right, but after he opened one of the doors and then offered me the choice, it’s now 66.7% if I choose to switch. So yeah I take door number two and thank you for that extra 33.3%.”

The problem is based on the television game show Let’s Make a Deal and named after its original host, Monty Hall.

Draw a tree diagram and see WHAT happened

Generally speaking, when you see the host specially open a door and give you a chance to switch your choice, most of people would think that host is playing a trick on them and insists to stay their choice. Or be more rational, the probability of picking a car should be 50% intuitively. Why the probability is 66.7% if we switch the choice? The first step to learn Probability must learn how to draw a tree diagram first. Let’s draw all the condition and see what happened!

monty-hall-problem-01

We can easily get the probability which:

win the car after we switch the door: 
\(\displaystyle{= \frac{\frac{1}{6}+\frac{1}{6}}{\frac{1}{12}+\frac{1}{12}+\frac{1}{6}+\frac{1}{6}}}\)

\(\displaystyle{=\frac{\frac{1}{3}}{\frac{1}{2}}=\frac{2}{3}}\)

win the car after we stay at the same door:
\(\displaystyle{=\frac{\frac{1}{12}+\frac{1}{12}}{\frac{1}{12}+\frac{1}{12}+\frac{1}{6}+\frac{1}{6}}}\)

\(\displaystyle{=\frac{\frac{1}{6}}{\frac{1}{2}}=\frac{1}{3}}\)

See an extreme example to break the intuition

Suppose there are 1000 doors for you to choose. Now you choose a door and the host open other 998 doors ( remember the host know what’s behind the door. So what behind the opened door are all goats.) Would you switch your choice? 

You perhaps think that the host plays a trick on you, so you insist your choice. However, from the perspective of statistics, the probability of wining this game by the door you choose at first is 1/1000. Then the host must intentionally not pick the door with car (unless you have picked that car). He has picked the other 997 doors that won’t win and this action increase your chance of wining!

monty-hall-problem-02

Of course, it does not mean that you will win the prize after changing your choice, but the chance of getting the car after changing the door will greatly increase.

What about 1/2?

Still think that the probability 1/2 make sense? Well, I have the same problemXD It seems that we can’t go to the team of Mickey to play cards.

So, is there any explaination to the number 1/2? Yes, let me show you.

monty-hall-problem-03

This is the part of tree diagram we see previously. Do you notice that each probability in the last stage is 1/2?

That is, base on the problem of “to choose or note”, 1/2 is a reasonable answer. But! you ignore the previous condition.

This is also a fallacy that often occurs in daily life. Once some conditions happen, the probability of the event would change dramatically. Like Ben said, “It changed everything.”

Alex’s Choice

Chancing It: The Laws of Chance and How They Can Work for You

Author: Robert Matthews

Make your own luck by understanding probability

Over the years, some very smart people have thought they understood the rules of chance?only to fail dismally. Whether you call it probability, risk, or uncertainty, the workings of chance often defy common sense. Fortunately, advances in math and science have revealed the laws of chance, and understanding those laws can help in your everyday life.

Add some ingredients

♦ 蒙提霍爾問題(一)決勝21點
♦ Wikipedia – Monty Hall problem

Seasoned with Chinese

◊ Monty Hall Problem – 蒙提霍爾問題
◊ probability – 機率
◊ conditional probability – 條件機率
◊ tree diagram – 樹狀圖


Picture Credit:
the door in the image is from Flaticon (author: prettycons
the goat of the article is from Flaticon (author: Freepik)

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