Would You Switch?: A Look at the Curtain Game

In summary, the "guess which one of three curtains the prize is behind" scenario is a classic puzzle that many people get wrong. The correct answer is to switch to the third curtain after the second curtain is shown to be empty. This can be explained mathematically, through trial and error, and by exaggerating the example with a larger number of curtains. It is important to remember that the host's actions do not provide any new information about your initial choice, and therefore the odds are not evenly distributed after one curtain is revealed to be empty. This has been a well-known problem for many years and has sparked discussions and debates among mathematicians.
  • #1
jeff
Science Advisor
658
1
Consider the standard "guess which one of three curtains the prize is behind" scenario. After choosing curtain 1, you're shown that nothing resides behind curtain 2. If allowed, would you switch to curtain 3?
 
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  • #2
Of course!

- Warren
 
  • #3
Originally posted by chroot
Of course!

- Warren

I'm not surprised that you'd find this puzzle trivial, but you'd be surprised how many people get it wrong, and even when you explain it to them they don't believe it.

Btw, remember that dumb stanford bookstore joke? I think the mixup occurred because you'd made the comment about the stanford bookstore when my user name was steinitz, but just before I teased you, I'd had it changed to my given name "jeff" because it was annoying being called by a last name and obviously that's not something you'd have noticed.
 
  • #4
First of all, my answer is just like chroot's.
Here is the way i explain it :
1-Mathematically:
The chance of getting the right curtain from the first try is 1/3, after the second curtain is opened the chance of the prize being behin the third curtain will become 2/3, so i would choose the biggest chance.
2-Trial and error:
Write all the possibilites, you will see that if you change your mind about your curtain decision, you chance of winning will be the double of if you didn't (note that (1/3)*2=2/3 )
3-Exagerating:
Make the example bigger, suppose there are 100 curtains, and you choose 1 curtain, and 98 curtains are opened without the price behind them.
Now, the only curtain that was not opened and was not chosen by you was not opeend for a good reason, which is that the prize could be behind it.
There are two posibilited, either that the curtain which was not opened was not opened because the prize is behind it (among the 99 other ones), or to fool you because you had the right curtain from the first place !
Now the chance of you having the right curtain from the first place is only 1/100, so 99/100 is in favour of "it was not opened because the prize is behind it", so change your choice to the 99/100 !
 
  • #5
This is a very old (and very simple) problem. Even before it raised such a fuss in Marilyn Vos Savant's newspaper column a number of years ago, I found it as an exercise, in chapter 1, of an elementary probability text written 20 years ago.
 
  • #6
Originally posted by STAii

1-Mathematically:
The chance of getting the right curtain from the first try is 1/3, after the second curtain is opened the chance of the prize being behin the third curtain will become 2/3, so i would choose the biggest chance.

I get your last explanation... but for this one...


I don't see why it wouldn't be 50/50 between the 1st and 3rd.

Before: 1/3 ------ 1/3 ------ 1/3


After: 1/2 ------- 0 -------- 1/2


You say its: 1/3 ---- 0 ------ 2/3



Originally posted by STAii
2-Trial and error:
Write all the possibilites, you will see that if you change your mind about your curtain decision, you chance of winning will be the double of if you didn't (note that (1/3)*2=2/3 )

Why are you multiplying by 2?
The 2nd curtain's chance is now 0. Its chances are spread equally between the other 2, so we add .5(1/3) to the 1st and 3rd curtains. Where do you get the "multiply by 2" thing?


=============

Some more explanation is needed, please. (details!)


P.S. can someone please give the stanford bookstore joke?

edit: typed a wrong number
 
  • #7
I don't see why it wouldn't be 50/50 between the 1st and 3rd.

Because there's no longer a uniform distribution of probability.


In the beginning you see three curtains. You have no a-priori reason to think any is more likely than any other, so each of them has a one in three chance of having the prize.

So you select a curtain. The odds are one in three that you were correct, and two in three that you were incorrect.


Now the host opens one of the curtains to reveal there is no prize behind it.

Does that give you any new information about your selection? Nope! You all ready knew that at least one of the curtains you did not select did not hide the prize.



Here's another way of thinking of it. If the host had the freedom to open any of the three curtains he wanted, you'd then have a 50/50 chance when choosing which curtain you want... but if you tell the host "You cannot open curtain A", what then are the odds that the prize is behind curtain A or behind the other curtain he did not open?
 

1. What is the concept of the Curtain Game?

The Curtain Game, also known as the Monty Hall problem, is a probability puzzle that involves choosing between three curtains, behind which are a prize, a goat, and an empty space. The player makes an initial choice, after which the host reveals one of the other two curtains, always showing a goat. The player is then given the option to switch their choice or stay with their original one.

2. What is the likelihood of winning if I switch my choice?

Many people assume that the chances of winning are equal whether they switch their choice or not, but this is not the case. In fact, the chances of winning increase from 1/3 to 2/3 if the player switches their choice. This can be mathematically proven using conditional probability.

3. Is the Curtain Game a real-life scenario or just a theoretical puzzle?

The Curtain Game can be seen as a simplified version of real-life scenarios, such as choosing between different job offers or selecting a door in a game show. The concept of conditional probability and the idea of making strategic choices based on new information can be applied to various situations in real life.

4. Why is the Curtain Game so counterintuitive?

The reason the Curtain Game may seem counterintuitive is because our brains tend to rely on intuition and past experiences when making decisions. In this case, our intuition tells us that the chances of winning remain the same regardless of switching, but the logical and mathematical explanation proves otherwise.

5. How is the Curtain Game relevant in the field of science?

The Curtain Game is a popular example used in the study of probability and decision making. It highlights the importance of understanding conditional probability and making informed choices based on new information. This concept is valuable in various scientific fields, such as statistics, psychology, and economics.

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