This is a good puzzle - except I can't solve it!

[Stormrunner]

Another door variant:
A game show host shows you three identical doors, and informs you that one leads to a prize and the other two cause you to instantly lose all money you have won on the show so far. He asks you to pick one. After you pick, he opens one of the other two doors, revealing a "loser" sign. He then offers you the chance to change your pick, if you desire.
Are you better off sticking with your first pick, or choosing the other door?
[edit: you don't actually get to open the door until after your second pick, so when you make the second pick you still only know the contents of one door, the one the host opened.]

Example: you pick door C, he reveals that door B is a "bad" door. Do you pick door A or C?

Hint: the host knows what's behind each door, so the door he opens is never the one with the prize.

Hint 2: You cannot be 100% certain with this one, the question is how can you maximize your chances.


With the dotted monk puzzle, how do the monks know there are exactly seven dots? The examples all have monks who are all dotted. Can you give an example for, say, five monks, three of which are dotted? (So each monk can see at least one dotted and one non-dotted companion.)

 

[Lonely Tylenol]

Three doors

I've seen this one many times before (it's known as the Let's Make A Deal problem or the Monty Hall problem because that situation was a daily occurance on that particular game show hosted by that particular personality) and the explanation for the solution is actually kind of odd.

Spoiler

Long story short: When you choose from three doors, you have a 33.3% chance of picking the prize door. When one door is eliminated and you have a chance to pick a second time, you still have a 33.3% chance of success on the first door, because that was the probability when you picked it. But the chance of success for all doors must add up to 100% (or else there's a chance that no door has the prize). Since one door was eliminated that means that the chance of success for the unpicked door is 66.6% because 66.6% + 33.3% ~= 100%.

So in every case, your best odds are to change to the other door. It actually doubles your chance of winning. It sounds kind of wacky, but it works. Here's some more info on the problem. There are also websites with applets that will run the game for you to show empirically that the numbers work out this way.

 

[Gorilla726]

Well, as the beginning question has been answered many times over, I feel it is my duty to give an answer that I would allow my players to use.

"I ignore the two doors and promptly find me a stairway to Heaven!"

Any thoughts? Hehehe...

Gorilla

 

[Hypersmurf]

With the dotted monk puzzle, how do the monks know there are exactly seven dots? The examples all have monks who are all dotted. Can you give an example for, say, five monks, three of which are dotted? (So each monk can see at least one dotted and one non-dotted companion.)

No, you missed the point. There are sixty monks. Seven of them are dotted, but they are not told how many. (All they know is that there must be at least one dotted.)

The first six times the bell rings, none of the monks do anything. On the seventh ring, all seven dotted monks raise their hands.

In the example you give, the bell would ring once... no reaction. Twice... no reaction. Three times... and the three dotted monks raise their hands.

-Hyp.

 

[Hypersmurf]

I've seen this one many times before...

Quite recently, in fact, in this thread in the Off-Topic Forum.

-Hyp.

 

[jmucchiello]

"There are two guards in a corridor, one guarding a doorway to heaven, the other guarding a doorway to hell. One of the guards will always tell the truth but the other will always lie.
By asking a single question that both the guards answer, find the doorway to heaven."

What stops me from opening both doors and deciding from the two views? Nothing in the puzzle says the guards will not allow the doors to be opened.

 

[Hypersmurf]

What stops me from opening both doors and deciding from the two views?

It's not a solution to the puzzle. The puzzle asks you to determine the door by asking a single question.

Opening the doors might tell you which door is which, but it doesn't solve the puzzle.

-Hyp.

 

[Incenjucar]

The real answer to any of these is "Tell me the truth or I go epic on your arse!".

Heck, they had the riddle on that Yu-Gi-Oh cartoon, and just had the people both lying.

 

[Lonely Tylenol]

Quite recently, in fact, in this thread in the Off-Topic Forum.

That's a good thread!

 

[Asmor]

I've seen this one many times before (it's known as the Let's Make A Deal problem or the Monty Hall problem because that situation was a daily occurance on that particular game show hosted by that particular personality) and the explanation for the solution is actually kind of odd.

Spoiler

Long story short: When you choose from three doors, you have a 33.3% chance of picking the prize door. When one door is eliminated and you have a chance to pick a second time, you still have a 33.3% chance of success on the first door, because that was the probability when you picked it. But the chance of success for all doors must add up to 100% (or else there's a chance that no door has the prize). Since one door was eliminated that means that the chance of success for the unpicked door is 66.6% because 66.6% + 33.3% ~= 100%.

So in every case, your best odds are to change to the other door. It actually doubles your chance of winning. It sounds kind of wacky, but it works. Here's some more info on the problem. There are also websites with applets that will run the game for you to show empirically that the numbers work out this way.

I believe that to be flawed reasoning. Here's why:

Spoiler

If the host always, no matter what, opens a wrong door after the guest chooses and then gives the guest the chance to change his mind, then the first guess is entirely irrelevent. It is, for all intents and purposes, a choice between 2 doors, not 3, as you can be 100% sure that one of the wrong choices will be eliminated before you have to make your final decision. It is therefore a simple 50/50 chance.

Of course, I could be wrong. That's just how I see it. I hate to admit it, the mathematical proof does seem to make sense, but I don't have a strong enough background in whatever area of mathematics is relevent to the problem to decide how accurate the proof is, and it's a simple matter to make a false proof that will pass muster to the untrained eye.