Let's Make A Deal: Should You Switch Curtain Choices?

  • Thread starter JohnDubYa
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In summary: For example, the game show host might know that the tickets are behind curtain #1. In that case, you have no chance of winning unless you switch to curtain #1.
  • #1
JohnDubYa
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I hope this problem isn't old hat here. I know the answer, but I would be curious to see your explanations.



Suppose you are on a game show, where you are presented three curtains. Donkeys reside behind two of the curtains, and the big prize (tickets to Metallica) are behind the third curtain.

Suppose you pick curtain #1. Rather than show you what is behind curtain #1, the game show host opens curtain #2 to show you a donkey. Obviously the Metallica tickets must be behind curtain #1 (your original choice) or curtain #3.

Before opening curtain #1, the game show host decides to let you change your mind and pick #3 instead if you wish.

Should you switch to curtain #3, or stick by your original choice?

Or does it matter?

And why?
 
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  • #2
A note to everyone When I searched for this topic on the internet google searching "Monty Hall" turned up good results. It pointed out all though the question assumes Monty is not trying to trick the player in reality sometimes Monty did because he was giving to many cars away.

I think I have a better way of exaplaining why you should switch doors. First of all pretend their are 99 donkeys and one good prize rather than 2 donkeys and one good prize. Now imagine you have picked one door and the host and the lovely assistant are blocking the door you picked and one other door. when you are deciding wether to pick the door you realize that you can see the shadow of donkey hooves under 98 doors, this information would have been great to use before you picked and if you switch you are actually still making use of this information.
I still find this explaining this question also perplexs me, I fully understand how to calculate the actual probability but the probability the final probability seems dodgy unless . I find I am quite good at explaing probability questions (i Have tutored uni probability for 5 years. And I understand why some probabilities soudn strange to students. For example there is a simple reason why people get the following question wrong : "what is the chance that none of the students in a class of twenty has a common birthday". People interept the question as "what is the chance that I personally would have a common birthday with at least one person in a group of 20 "
 
  • #3
Are you absolutely certain that "Metallica" tickets are a better prize than donkeys?

The crucial point here is that the game show host knows which curtain the tickets are behind: he has knowledge you don't. As far as you know to start with, the tickets could be behind anyone of the three curtains: your chances are 1 in 3.

Once the game show host opens the curtain revealing the donkeys (and, of course, he wouldn't open the curtain the tickets are behind), you now have knowledge you did not have before. There was a 1/3 chance that you chose the right curtain before. In that case, the host could open either of the two remaining curtains. There was a 2/3 chance that you chose a wrong curtain, in which case the host MUST open a wrong curtain, leaving the "right" curtain closed. Since there is a 2/3 chance of that happening, as opposed to a 1/3 chance of the other way, yes, you can improve your chances of winning from 1/3 to 1/2 by changing your choice.

"Marilyn Vos Savant" gave a very clear version of this several years ago: Suppose there were one million curtains with donkeys behind them and 1 curtain with the prize. You chose one of the 1000001 curtains, then the host removes all but your choice and one other! Since you had a 1 in 1000001 chance of choosing the correct curtain to start with isn't it clear that you had better change to the one left?

By the way, it is an interesting calculation to see what happens if Monty Hall (oops, "the host") did not know which curtain the prize was behind.

Suppose there are three curtains- you choose one. The host chooses one of the remaining two to open, at random but it happens not to be the prize. You can show that, in this case, using the conditional probability that host will not choose the correct curtain assuming you also did not, that whether you change or not, your probability of winning remains 1/3. No additional knowledge, no improved chances!
 
  • #4
You start with a 1/3 chance of success. IWhen one curtain is revealed to be a failure the other 2 curtains have 1/2 cance of success, so it does not matter weather you change your selection. This is providing that you have no other knowledge such as hearing the donkey!
 
  • #5
RE: "
Suppose there are three curtains- you choose one. The host chooses one of the remaining two to open, at random but it happens not to be the prize. You can show that, in this case, using the conditional probability that host will not choose the correct curtain assuming you also did not, that whether you change or not, your probability of winning remains 1/3. No additional knowledge, no improved chances!"

I don't agree. The game show host may not have any knowledge of the donkeys, but you do now that the curtain has been opened. I don't think it matters if the game show host knew which donkey the curtain was behind.
 
  • #6
RE: "IWhen one curtain is revealed to be a failure the other 2 curtains have 1/2 cance of success, so it does not matter weather you change your selection."

I definitely don't agree, and I think this can be disproven with a simple computer algorithm. In fact, this would be a fun programming problem at the introductory level.
 
  • #7
RE: "Are you absolutely certain that "Metallica" tickets are a better prize than donkeys?"

Yes, unless you are Napster.
 
  • #8
HallsofIvy said:
Once the game show host opens the curtain revealing the donkeys (and, of course, he wouldn't open the curtain the tickets are behind), you now have knowledge you did not have before. There was a 1/3 chance that you chose the right curtain before. In that case, the host could open either of the two remaining curtains. There was a 2/3 chance that you chose a wrong curtain, in which case the host MUST open a wrong curtain, leaving the "right" curtain closed. Since there is a 2/3 chance of that happening, as opposed to a 1/3 chance of the other way, yes, you can improve your chances of winning from 1/3 to 1/2 by changing your choice.
You actually improve your chances from 1/3 to 2/3 by changing your choice.

If you don't switch, your chances of winning remain at 1/3. If you flip a coin to decide which remaining curtain to choose, your odds rise to 1/2. But if you take full advantage of the new information given you--and always change your choice--you will increase your odds to 2/3.

Using Marilyn's example of 1,000,001 curtains, the odds are overwhelming that your first pick is wrong (1/1,000,001) and that the other choice is right (1,000,000/1,000,001). In this case, changing your choice is almost a sure thing!
 
  • #9
If the Host chooses a curtain at random I'd say then that as long as it's a donkey he exposes then you have only changed the odds to 50/50. I actually had an argument with a friend of mine over this particular question a while back. It comes down to the added information that the Host will not reveal the prize because he has foreknowledge of it's location. Due to that then by changing your choice your probability of having chosen the proper curtain increases to 2/3 instead of 1/3. And my friend actually did write a program that illustrated the point.
 
  • #10
JohnDubYa said:
RE: "
Suppose there are three curtains- you choose one. The host chooses one of the remaining two to open, at random but it happens not to be the prize. You can show that, in this case, using the conditional probability that host will not choose the correct curtain assuming you also did not, that whether you change or not, your probability of winning remains 1/3. No additional knowledge, no improved chances!"

I don't agree. The game show host may not have any knowledge of the donkeys, but you do now that the curtain has been opened. I don't think it matters if the game show host knew which donkey the curtain was behind.

You choose one of the three curtains- there is an a-priori probability of 1/3 that you have choose the right curtain.
Suppose the game show host does NOT know (or has forgotten) which curtain has the prize but opens one of the remaining two "at random" (the two being equally likely). Of course, if he opens the curtain that DOES have the prize, all bets are 0ff- and next week the show has a new host- so we must look at the conditional probability given that he DOES choose the curtain that does not have the prize.

IF the curtain you chose was the one with the prize (there was a 1/3 prob of that) then it doesn't matter which curtain the host chooses, it will NOT have the prize behind it and if you switch, you lose.

IF the curtain you chose was not the one with the prize (there was a 2/3 probability of that) then there is also a 1/2 probability that the host will also choose a curtain without the prize so that the show CAN "go on": there is a (1/2)(2/3)= 1/3 probability that this will happen. Now, if you switch, you win.

Notice that there is a 1/3 probability that switching will win and a 1/3 probability that switching will lose: your probability is exactly the same as if you ignored the host and kept the same curtain. (Of course, now there is also that 1/3 probability that the host will open a curtain that reveals the prize and ends the show!)

That is: IF the host knows which curtain the prize is behind and intentionally opens a curtain that does NOT have the prize (the way the show is supposed to work) then switching curtains will raise your probability of winning from 1/3 to 1/2. IF the host does NOT know which curtain the prize is behind and by sheer luck opens a curtain that does NOT have the prize, swiching curtains DOES not change your probability of winning.

(Personally, I think donkeys would sound better than metallica!)
 
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  • #11
HallsofIvy said:
That is: IF the host knows which curtain the prize is behind and intentionally opens a curtain that does NOT have the prize (the way the show is supposed to work) then switching curtains will raise your probability of winning from 1/3 to 1/2.
No, the probability of winning goes from 1/3 to 2/3. It doubles.
IF the host does NOT know which curtain the prize is behind and by sheer luck opens a curtain that does NOT have the prize, swiching curtains DOES not change your probability of winning.
Sure it does. I don't care what the host knows or doesn't know. If he opens a curtain that does not have the prize (whether by blind luck or on purpose) you are being given new information. Switch curtains and you double your chances of winning.
(Personally, I think donkeys would sound better than metallica!)
I'm with you on that one! :smile:
 
  • #12
A good way to look at this.

Pick a door any door (or is it a curtian? does it matter? Sure, door is easier to spell!)
You have a 1/3 chance of picking the winning door. That means that there is a 2/3 chance that the prize is behind one of the remaining 2 doors. The host shows you the door that it is NOT behind. By doing this he has not changed the odds, but he has eliminated the uncertainty. The odds remain 2/3 that the prize is behind the remaining door.

When I first encountered this problem I was skeptical, so I wrote a simple VB program to guess a 1000 times. The results were exactly 2/3.
 
  • #13
it seems to me that 2/3 chance would mean that 2 out of 3 doors contained the prize. since the prize is only behind one door out of two(at the middle of the game) then why wouldn't the chances be 1/2? It seems to me that at first there is a 1/3 chance of picking the right door, then after a losing door is removed leaving only two choices (one being the prize), then that would convert the odds to 1/2. No matter which door you chose to begin with, you are still making a definate choice to either stick with the original door or switch, both of witch seem to me to have a 1/2 chance of winning after one of the losing doors has been removed. This is how it makes sense to me, but as i have read above i might not know what I am talking about.
 
  • #14
you need to know about conditional probability, and to read the question carefully. it is important whether or not the host knows in advance where the prize is, and is thus guaranteed to open a non-prize door. if so then you must change, if not then it doesn't matter
 
  • #15
Could someone explain why it matters whether the host knows in advance or not? If the host picks a door that doesn't have the prize, 2/3 of the time you will select the prize by switching. It seems that it doesn't matter whether the host knows where the prize is.
 
  • #16
Agreed. Every logical analysis of this game that I have seen is unaffected by whether or not Monty knows which curtain has the Metallica tickets.
 
  • #17
Monty's ignorance will cost you

JohnDubYa said:
Agreed. Every logical analysis of this game that I have seen is unaffected by whether or not Monty knows which curtain has the Metallica tickets.
Of course it matters whether Monty knows where the prize is!

If Monty doesn't know, then the game will work a bit differently. 1/3 of the time he would reveal the prize! Well, no point in switching then--you know you lost. 2/3 of the time, he would--again, by chance alone--reveal an empty curtain. In that case, you'd better switch! Your odds improve from 1/3 to 2/3 for that case where an empty curtain is revealed.

In the original game, in which Monty always reveals an empty curtain (because he knows where everything is), your chances of winning are 2/3 by using the strategy of always switching to the remaining curtain.

In this new variation, in which Monty knows nothing, your odds of winning are 2/3 * 2/3 = 4/9 if you use the strategy of always switching to the remaining curtain whenever Monty reveals an empty curtain.
 
  • #18
I think you have a fault in your logic. When you multiplied the two 2/3's, you incorporated both cases where Monty picks the curtain that has the Metallica tickets, as well as the case where Monty picks the curtain with the donkey. But we can throw out the former, because the situation only calls for those instances where Monty reveals an empty curtain.

Suppose Monty doesn't know. You pick curtain #1, and he decides (on a lark) to open curtain #3, showing a donkey.

Do you switch?

To me, what matters is the information YOU gain, no matter how that information was gained. You were 2/3 likely to be wrong with your initial guess, and that hasn't changed.
 
  • #19
JohnDubYa said:
I think you have a fault in your logic. When you multiplied the two 2/3's, you incorporated both cases where Monty picks the curtain that has the Metallica tickets, as well as the case where Monty picks the curtain with the donkey. But we can throw out the former, because the situation only calls for those instances where Monty reveals an empty curtain.
But I do throw out the former. That's why it's 2/3 * 2/3 and not 3/3 * 2/3.
Suppose Monty doesn't know. You pick curtain #1, and he decides (on a lark) to open curtain #3, showing a donkey.

Do you switch?
Absolutely! Once I get to that point, I can make my odds of winning 2/3 by switching. (Of course, I only have a 2/3 chance of getting to that point.)
To me, what matters is the information YOU gain, no matter how that information was gained. You were 2/3 likely to be wrong with your initial guess, and that hasn't changed.
Right. And you gain useable information only when Monty reveals a donkey. Which doesn't happen quite as often in your new (and quite different!) scenario.
 
  • #20
Everyone who believes there is only a 1/2 chance needs to read (or preferably listen to the audiobook of):

The Curious Incident of the Dog in the Night-Time

Not only is it an excellent book, it references this paradox
(yes, it IS a paradox)
And CLEARLY explains why it is favorable to switch, based on statistics.
 
  • #21
Or for those of you who don't want to go through the bother of being well-entertained by an excellent story:

There are 3 possible outcomes:
Donkey 1, Donkey 2, or Tickets

Let's say you pick one of the curtains, you then have a 1/3 chance of having chosen either the Donkey 1, Donkey 2, or Tickets.

Without knowing what is behind the curtain you've chosen, the announcer reveals A donkey behind one of the OTHER curtains and asks if you would like to switch:

-If Donkey 1 is behind your curtain, and the announcer reveals Donkey 2, if you switch, you get the Tickets.
-If Donkey 2 is behind your curtain, and the announcer reveals Donkey 1, if you switch, you get the Tickets.
-If the Tickets are behind your curtain, and the announcer reveals one of the donkeys and you switch, you get the other donkey.

Mystery solved. No more trolling now.
 
  • #22
Also, just a note for the OP, if you really want to see this process in action use a statistical package and create the right random variables and execute this process a few thousands times and then see what happens.

If you every do further probability or statistics, this is a good way to help you correct your intuition if it is not on the right path.
 

1. How does "Let's Make A Deal" work?

"Let's Make A Deal" is a game show where contestants are presented with a choice between three curtains. Behind one curtain is a valuable prize, while the other two curtains have less desirable prizes or no prize at all. The contestant chooses a curtain, and then the host offers them the option to switch to a different curtain. The contestant must decide whether to keep their original choice or switch to the other curtain.

2. Why is it called the "Monty Hall Problem"?

The name "Monty Hall Problem" comes from the original host of "Let's Make A Deal," Monty Hall. He would often offer contestants the option to switch curtains, leading to the famous dilemma that has become known as the "Monty Hall Problem."

3. What is the probability of winning if you switch curtains?

The probability of winning if you switch curtains is 2/3, or approximately 66.67%. This is because when you initially choose a curtain, you have a 1/3 chance of choosing the correct one. However, when given the option to switch, the remaining 2/3 probability is split between the other two curtains, making the probability of winning if you switch 2/3.

4. Is it always better to switch curtains?

In most cases, it is better to switch curtains. As mentioned before, the probability of winning if you switch is 2/3, while the probability of winning if you stay with your original choice is only 1/3. However, this is only true if the host knows which curtain contains the prize and always offers the option to switch. If the host does not know which curtain has the prize or does not always offer the option to switch, the probabilities may vary.

5. Why do people often choose to stay with their original curtain choice?

People often choose to stay with their original curtain choice because they believe that their initial decision was correct and they do not want to risk losing the prize. Additionally, they may feel that switching is based on luck and not a strategic decision. However, as shown by the probability, switching curtains actually increases the chances of winning.

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