This is by now a venerable chestnut but may tickle the brain cells of those who have not yet encountered it.
Here goes:
You are at an aerodrome with a mystery benefactor and in front of you are three identical planes, the only difference being that they are respectively marked A, B and C . You are invited to choose to travel on one plane of your choice. Two of the three planes will take you for a day trip to Denver. The other plane will take you to Mauritius for a four week all expenses paid holiday at a six star hotel. There is no way of physically identifying which plane is travelling where. You decide to pick plane A. Immediately afterwards, your benefactor says before you confirm your decision I'm now going to tell you that Plane B is going to Denver. Should you stick with Plane aA or switch to Plane C.?
Switch.
Pick plane B and come visit me!
Quote from: Ali on May 07, 2012, 09:06:17 PM
Pick plane B and come visit me!
Has Ikea got its own landing strip?
Switch for greatly improved chances.
Quote from: En_Route on May 07, 2012, 09:21:59 PM
Quote from: Ali on May 07, 2012, 09:06:17 PM
Pick plane B and come visit me!
Has Ikea got its own landing strip?
Yes, but you have to build it yourself.
Quote from: Ali on May 07, 2012, 09:06:17 PM
Pick plane B and come visit me!
I'd switch to B in a heartbeat, I'd much sooner hang out with you any day.
^^ D'awwwwwwwwwwww!
Quote from: En_Route on May 07, 2012, 08:39:41 PM
This is by now a venerable chestnut but may tickle the brain cells of those who have not yet encountered it.
Here goes:
You are at an aerodrome with a mystery benefactor and in front of you are three identical planes, the only difference being that they are respectively marked A, B and C . You are invited to choose to travel on one plane of your choice. Two of the three planes will take you for a day trip to Denver. The other plane will take you to Mauritius for a four week all expenses paid holiday at a six star hotel. There is no way of physically identifying which plane is travelling where. You decide to pick plane A. Immediately afterwards, your benefactor says before you confirm your decision I'm now going to tell you that Plane B is going to Denver. Should you stick with Plane aA or switch to Plane C.?
It depends on the reason (the rule) behind the benefactor picking plane B to tell you it goes to Denver.
Benefactor method (Random selection)If the rule was random selection then it doesn't matter which one you choose, whether you stay or whether you switch, your chances are 50/50
because both A and C had originally 33.3333% chance of being Mauritius.
The benefactor had a 66.6666% chance that B was Denver, and as it turned out this is what B was. (the important thing to note is that the one the benefactor chose, could have been Mauritius (assuming the method was random selection)
Benefactor method (always picks Denver)If the rule was that the benefactor always selects "Denver" because the benefactor has prior knowledge of which plane goes where, then this means that
the benefactor specifically chose B because B
was going to "Denver", (the important thing to note is that the benefactor would not have chosen B if B went to Mauritius, instead would have picked C)
Why is this important?
With three options and only one Mauritius there is a 33.3333% chance that any one of them are going to Mauritius.
There is a 66.6666% chance that both B and C are either Denver and Mauritius respectively or Mauritius and Denver respectively but only a 33.3333% chance that both B and C are Denver this is because there is a 33.3333% chance that A is Mauritius.
So the benefactor must choose one Denver plane and reveal this to the contestant, it could either be B or C, there will always be at least one Denver plane within B and C. The one left over has a 66% chance of being Mauritius because 66.666% of the time one of the two planes would have had Mauritius.
So under this condition the contestant ought to swap as A has 33.3333% chance of being Mauritius where as C has 66.6666% chance.
Does this make sense?
Quote from: Stevil on May 07, 2012, 11:17:23 PM
Quote from: En_Route on May 07, 2012, 08:39:41 PM
This is by now a venerable chestnut but may tickle the brain cells of those who have not yet encountered it.
Here goes:
You are at an aerodrome with a mystery benefactor and in front of you are three identical planes, the only difference being that they are respectively marked A, B and C . You are invited to choose to travel on one plane of your choice. Two of the three planes will take you for a day trip to Denver. The other plane will take you to Mauritius for a four week all expenses paid holiday at a six star hotel. There is no way of physically identifying which plane is travelling where. You decide to pick plane A. Immediately afterwards, your benefactor says before you confirm your decision I'm now going to tell you that Plane B is going to Denver. Should you stick with Plane aA or switch to Plane C.?
It depends on the reason (the rule) behind the benefactor picking plane B to tell you it goes to Denver.
Benefactor method (Random selection)
If the rule was random selection then it doesn't matter which one you choose, whether you stay or whether you switch, your chances are 50/50
because both A and C had originally 33.3333% chance of being Mauritius.
The benefactor had a 66.6666% chance that B was Denver, and as it turned out this is what B was. (the important thing to note is that the one the benefactor chose, could have been Mauritius (assuming the method was random selection)
Benefactor method (always picks Denver)
If the rule was that the benefactor always selects "Denver" because the benefactor has prior knowledge of which plane goes where, then this means that the benefactor specifically chose B because B was going to "Denver", (the important thing to note is that the benefactor would not have chosen B if B went to Mauritius, instead would have picked C)
Why is this important?
With three options and only one Mauritius there is a 33.3333% chance that any one of them are going to Mauritius.
There is a 66.6666% chance that both B and C are either Denver and Mauritius respectively or Mauritius and Denver respectively but only a 33.3333% chance that both B and C are Denver this is because there is a 33.3333% chance that A is Mauritius.
So the benefactor must choose one Denver plane and reveal this to the contestant, it could either be B or C, there will always be at least one Denver plane within B and C. The one left over has a 66% chance of being Mauritius because 66.666% of the time one of the two planes would have had Mauritius.
So under this condition the contestant ought to swap as A has 33.3333% chance of being Mauritius where as C has 66.6666% chance.
Does this make sense?
Not at all. Although I concur that you're right because I've read that explanation several times in several different places in response to several different "3 doors" type questions. I still don't understand why it's not 50-50 between the two remianing doors (and suspect I never will understand it), but since I know that somehow, it's not, I know that if I am ever in that situation, I will switch.
It's easier to see if you reframe the problem. Imagine you have to pick the ace of spades from a full deck of cards. You pick a card face down, leaving 51 cards. At that point somebody tells you he will go through the remaining 51 cards and throw out 50 that are not the ace of spades. You then can elect either to hold on to your first pick or select the card that remains after the cull of 50 cards.
...Making your first bet at a 1/52 chance and the second a near-sure-win if you switch.
Quote from: Ali on May 07, 2012, 11:21:54 PM
Not at all. Although I concur that you're right because I've read that explanation several times in several different places in response to several different "3 doors" type questions. I still don't understand why it's not 50-50 between the two remianing doors (and suspect I never will understand it), but since I know that somehow, it's not, I know that if I am ever in that situation, I will switch.
the permutations of B and C are:
Perm 1
Plane B = Denver
Plane C = Denver
Perm 2
Plane B = Mauritius
Plane C = Denver
Perm 3
Plane B = Denver
Plane C = Mauritius
These are the only options available, there are 3 options hence each option has a 1 in 3 chance = 33.333%
Perm 2 and Perm 3 include Mauritius, so there is a 2 in 3 chance that one of the two planes has Mauritius = 66.666%.
So if the benefactor chooses a plane and shows that it goes to Denver then there is a 66.666% chance that the other plane goes to Mauritius.
Quote from: En_Route on May 07, 2012, 11:56:04 PM
It's easier to see if you reframe the problem. Imagine you have to pick the ace of spades from a full deck of cards. You pick a card face down, leaving 51 cards. At that point somebody tells you he will go through the remaining 51 cards and throw out 50 that are not the ace of spades. You then can elect either to hold on to your first pick or select the card that remains after the cull of 50 cards.
But again, the key is that the person filtering is following a rule. They will not through out the ace of spades if they find it.
Based on this rule the odds change.
So in the initial OP, if the benefactor did not have a rule then the chances are 33.333% plane A or 33.3333% plane C. It wouldn't matter if you swapped.
But if the benefactor is guaranteed to show a Denver plane then the odds change and plane C becomes 66.6666% likely to be Mauritius
(https://www.happyatheistforum.com/forum/proxy.php?request=http%3A%2F%2Fi.imgur.com%2FGVS7M.png&hash=68e9a2705e02bc032b6f2b59053aac21ef09ef69)
When should you not switch?
When benefactor has two duds.
How often does he have two duds?
2 out of 6 = a third of the time.
So you should switch 'cause 66.66% of the time it pays.
The reasoning assumes as Stevil says the benefactor always reveals a dud so the info doesn't change the odds.
Suppose somebody comes in at the halfway stage. He doesn't know anything about the third plane - all he sees are two planes, one going to Mauritius and one to Denver. If you ask that person the odds, he rightly says 50-50.
Why are the odds different if that person doesn't come?
Quote from: OldGit on May 08, 2012, 09:46:55 AM
Suppose somebody comes in at the halfway stage. He doesn't know anything about the third plane - all he sees are two planes, one going to Mauritius and one to Denver. If you ask that person the odds, he rightly says 50-50.
Why are the odds different if that person doesn't come?
The odds aren't different, this new guy just doesn't have sufficient information to calculate the odds.
It's a restating of the Monty Hall problem. The Mythbusters showed it is better to switch after the 'reveal' than it is to stay with your first choice. Don't ask me for links - I got off work at 2 am (local time), home at 3 am, and it's now 4:40 am and I've had two beers. It is time for bed. You know how to use Google. ;D
I still don't get it. ;D But I believe you.
Same here. I've read half a dozen explanations and I have to accept it, but I simply can't get past 2 planes/2 destinations/50-50. ;D
Quote from: OldGit on May 08, 2012, 07:18:14 PM
Same here. I've read half a dozen explanations and I have to accept it, but I simply can't get past 2 planes/2 destinations/50-50. ;D
Same.
Quote from: DeterminedJuliet on May 08, 2012, 07:57:51 PM
Quote from: OldGit on May 08, 2012, 07:18:14 PM
Same here. I've read half a dozen explanations and I have to accept it, but I simply can't get past 2 planes/2 destinations/50-50. ;D
Same.
Bear in mind it's not just any two planes. One of the planes is the remainder of a pair after somebody who knows which is the right plane has intervened; the choice is therefore no longer based on pure randomness.
Quote from: Ali on May 07, 2012, 10:14:25 PM
Quote from: En_Route on May 07, 2012, 09:21:59 PM
Quote from: Ali on May 07, 2012, 09:06:17 PM
Pick plane B and come visit me!
Has Ikea got its own landing strip?
Yes, but you have to build it yourself.
:D. Haven't laughed so much for years. In fact, come to think of it, I haven't laughed for years.
Quote from: En_Route on May 08, 2012, 08:23:49 PM
Quote from: Ali on May 07, 2012, 10:14:25 PM
Quote from: En_Route on May 07, 2012, 09:21:59 PM
Quote from: Ali on May 07, 2012, 09:06:17 PM
Pick plane B and come visit me!
Has Ikea got its own landing strip?
Yes, but you have to build it yourself.
:D. Haven't laughed so much for years. In fact, come to think of it, I haven't laughed for years.
Glad to have broken a dry spell. :D
Quote from: En_Route on May 08, 2012, 08:23:49 PM
Quote from: Ali on May 07, 2012, 10:14:25 PM
Quote from: En_Route on May 07, 2012, 09:21:59 PM
Quote from: Ali on May 07, 2012, 09:06:17 PM
Pick plane B and come visit me!
Has Ikea got its own landing strip?
Yes, but you have to build it yourself.
:D. Haven't laughed so much for years. In fact, come to think of it, I haven't laughed for years.
*Really* laughed..? I think I can beat your years - it's been over a decade here. Oh, I can manage some occasional polite laughter at a joke in a properly politically correct setting, but I don't find many things worth laughing about.
Eh... I suppose something has to put Grumpy in the Grumpy Lumpy.
Quote from: Asmodean on May 08, 2012, 09:52:59 PM
Quote from: En_Route on May 08, 2012, 08:23:49 PM
:D. Haven't laughed so much for years. In fact, come to think of it, I haven't laughed for years.
*Really* laughed..? I think I can beat your years - it's been over a decade here. Oh, I can manage some occasional polite laughter at a joke in a properly politically correct setting, but I don't find many things worth laughing about.
Eh... I suppose something has to put Grumpy in the Grumpy Lumpy.
You guys
are grumps. It's rare I go a whole day without at least a couple of serious belly laughs. I don't think my life is that much happier than either of yours, I just manage to find a lot of humor in the stuff around me.
...While me, I see tragedy in every comedy. That, or overanalyse. I like doing that.
Quote from: Asmodean on May 08, 2012, 09:52:59 PM
Quote from: En_Route on May 08, 2012, 08:23:49 PM
Quote from: Ali on May 07, 2012, 10:14:25 PM
Quote from: En_Route on May 07, 2012, 09:21:59 PM
Quote from: Ali on May 07, 2012, 09:06:17 PM
Pick plane B and come visit me!
Has Ikea got its own landing strip?
Yes, but you have to build it yourself.
:D. Haven't laughed so much for years. In fact, come to think of it, I haven't laughed for years.
*Really* laughed..? I think I can beat your years - it's been over a decade here. Oh, I can manage some occasional polite laughter at a joke in a properly politically correct setting, but I don't find many things worth laughing about.
Eh... I suppose something has to put Grumpy in the Grumpy Lumpy.
I have long suppresses any tendencies to public expressions of mirth as I do not wish my aura of gravitas to be compromised. Plus I have crooked teeth.
Quote from: Ali on May 08, 2012, 10:14:04 PM
Quote from: Asmodean on May 08, 2012, 09:52:59 PM
Quote from: En_Route on May 08, 2012, 08:23:49 PM
:D. Haven't laughed so much for years. In fact, come to think of it, I haven't laughed for years.
*Really* laughed..? I think I can beat your years - it's been over a decade here. Oh, I can manage some occasional polite laughter at a joke in a properly politically correct setting, but I don't find many things worth laughing about.
Eh... I suppose something has to put Grumpy in the Grumpy Lumpy.
You guys are grumps. It's rare I go a whole day without at least a couple of serious belly laughs. I don't think my life is that much happier than either of yours, I just manage to find a lot of humor in the stuff around me.
I'm prettymuch the same way you are, Ali. While I might not genuinely laugh every day, more often than not, I find the humour in the stuff around me. All kinds of crazy little things. It's hard not to crack up laughing sometimes! If I feel the need to be sad, then fine, I'll let myself be sad and moody for a while. But most of the time, I prefer to be goofy, to joke around, to find joy where I can bloody well find it. :D
Quote from: En_Route on May 08, 2012, 11:45:58 PM
I have long suppresses any tendencies to public expressions of mirth as I do not wish my aura of gravitas to be compromised. Plus I have crooked teeth.
Ah! The Asmo has enough gravitas for a cemetery... Or a black hole, so His aura can tolerate some minor blows. I do should get my teeth whitened though... :-\
I'm not saying anything on this topic. ;D
Quote from: Asmodean on May 09, 2012, 07:28:49 AM
Quote from: En_Route on May 08, 2012, 11:45:58 PM
I have long suppresses any tendencies to public expressions of mirth as I do not wish my aura of gravitas to be compromised. Plus I have crooked teeth.
Ah! The Asmo has enough gravitas for a cemetery... Or a black hole, so His aura can tolerate some minor blows. I do should get my teeth whitened though... :-\
It's OK, Asmo, keeping them grey helps the whole lumpy package blend in better. :D That way, they never see you sneaking up on them in the dark.