A logical exercise

[En_Route]

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.?

[xSilverPhinx]

Switch.

[Ali]

Pick plane B and come visit me!

[En_Route]

Pick plane B and come visit me!

Has Ikea got its own landing strip?

[Asmodean]

Switch for greatly improved chances.

[Ali]

Pick plane B and come visit me!

Has Ikea got its own landing strip?

Yes, but you have to build it yourself.

[Amicale]

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.

[Ali]

^^ D'awwwwwwwwwwww!

[Stevil]

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?

[Ali]

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.

[En_Route]

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.

[Asmodean]

...Making your first bet at a 1/52 chance and the second a near-sure-win if you switch.

[Stevil]

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.

[Stevil]

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

[The Magic Pudding]

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.