The Monty Hall Game Show Conundrum

Diamond Geezer

Diamond Geezer

Gone But Not Forgotten
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Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, nothing. You pick a door, say No. 1, and the host, who knows what's behind the other doors, opens another door, say No. 3, which has nothing behind it. He then says to you, 'Do you want to change your mind and pick door No. 2?'

Is it to your advantage to take the switch ?

This problem was given the name The Monty Hall Paradox in honor of the long time host of the television game show "Let's Make a Deal." Articles about the controversy appeared in the New York Times and other papers around the country. The answer was that the contestant should switch doors and instigated 10,000 responses from readers, most of them disagreeing. Several were from mathematicians and scientists whose responses ranged from hostility to disappointment at the nation's lack of mathematical skills.

This question seems to have a non-intuitive answer. Why were so many convinced that switching was wrong? They had all decided that it did not matter if the contestant switched or did not switch. There may be a reason so many disagreed. Omitting one phrase in the statement of this problem changes the answer completely and this might explain why many people have the wrong intuition about the solution. If the host (Monty Hall) does not know where the car is behind the other two doors, then the answer to the question is "IT DOESN'T MATTER IF THE CONTESTANT SWITCHES." The change in the statement of the problem is so slight that this might be the reason this problem is such a "paradox."

It would appear that those who switched doors won about 2/3 of the time and those who didn't switch won about 1/3 of the time. Why is there such a large difference? I mean, once Monty shows you what's behind one of the doors, there are only two doors left. Right? Right. And the car is behind one of those two doors with equal probability. Right or wrong?

In other words, you are twice as likely to win if you switch than if you don't switch

Aren't you ?
 
He's seperated the doors into two groups. Group A has one door (the one originally chosen by you the contestant). Group B has 2 doors.

There's 1 chance in 3 group A contains the car.
There's 2 chances in 3 group B contains the car.

Now look at the problem this way.

You can select either group A (1 chance in 3 it contains the car) or you can select group B (2 chances in 3 it contains the car). If you select group B then he'll effectively tell you which door it's behind by disclosing the empty one.

Thus you are twice as likely to win the car by discarding your original selection.
 
Not, Grasshoppers, until you remind me of the answer to this conundrum:

Two men guard the gates to Paradise (or some such scenario): one is sworn to tell the truth to the questions asked, the other is sworn to lie. In order to avoid going to Hades in a handbasket, you have to ask the guards if Paradise lies beyond their door.

I've forgotten the correct question, but is it something along the lines of "If I ask you if Paradise lies behind your door, will you be telling the truth if you lie?"

If the guard is the liar, and says he will be telling the truth if he's lying, then he's telling the truth, but if.... no, uh, hang on... just a mo... :brows: :brows:
 
Let's assume that the correct answer is A, the incorrect one B. You are allowed to ask only one question of one, but not bot, of the guards.

The question is this: "If I ask your colleague, what will he say?"

If you are addressing the question to the truth teller, he will say "B".
If you are asking the liar, he too will say "B".

The answer is, therefore, "A".
 
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