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Monty and the Captive Goats

The Monty Hall Dilemma is a regular feature on weblogs, but after it (again) triggered lengthy discussions at work the other day I can’t resist trying to set things off elsewhere. Once you’ve got your head round it, try mentioning it to your colleagues and see how many fervently dispute the answer.

Wikipedia has a thorough analysis of the probabilities, and it’s covered in The Curious Incident of the Dog in the Night-time (which I really should read sometime).


Comments

I read the problem and solution, then decided to try to work it out on my own. This is what I came up with:

Scenario 1: If Monty is picking deliberately when possible...
And you picked the car (33%), so Monty picked randomly.
Odds of getting the car if you switch: 0%
Odds of getting the car if you don't switch: 100%

Scenario 2: If Monty is picking deliberately when possible...
And you picked the goat (66%), so Monty picked a goat deliberately.
Odds of getting the car if you switch: 100%
Odds of getting the car if you don't switch: 0%

Scenario 3: If Monty is picking randomly...
And you picked the car (33%), and Monty picked a goat randomly.
Odds of getting the car if you switch: 0%
Odds of getting the car if you don't switch: 100%

Scenario 4: If Monty is picking randomly...
And you picked the goat (66%), and Monty picked a goat randomly.
Odds of getting the car if you switch: 50%
Odds of getting the car if you don't switch: 0%

Scenarios 1 and 2: If you switch, 33% chance of failure, 66% chance of success
Scenarios 1 and 2: If you don't, 66% chance of failure, 33% chance of success
Scenarios 3 and 4: If you switch, 66% chance of failure, 33% chance of success
Scenarios 3 and 4: If you don't, 66% chance of failure, 33% chance of success

If you switch, your odds will either be 33% or 66% (though you don't know the likelihood of each)
If you don't switch, your odds are always 33%

It is always potentially better to switch.

— Greg, 25th Apr, 11:14pm

Your conclusion to switch is correct, but I don't quite understand your working...

Monty always picks a goat - so you always have a 1 in 3 chance if you stick, and a 2 in 3 chance if you switch.

— G. Herder, 26th Apr, 10:26am

Monty has to leave the winning door alone, so in the 2 out of 3 situations where you don't pick the winner the first time, he's always going to leave a winning door for you to switch to.

— morcs, 26th Apr, 4:00pm

The reason that your odds may not be 2/3 if you switch is because you do not know for a fact that Monty is picking the goat deliberately. He may just have picked randomly, in which case your odds are still 1/3. Since you can't calculate the odds of Monty picking deliberately or randomly, you have no idea if your odds are 1/3 or 2/3. However, if you don't switch, your odds are 1/3 no matter what.

— Greg, 26th Apr, 7:56pm

The wording used to introduce this problem is often slightly ambiguous, but the idea is that Monty always deliberately chooses a goat, rather than it just happening to be a randomly-selected goat in that particular case (in which case, yes, there'd be no advantage if switching).

So far morcs gets the award for shortest clear explanation of the solution.

Matt Round, 26th Apr, 10:02pm

Ahh, yes, with that interpretation of the problem it is a lot simpler, and a lot clearer that you should switch.

— Greg, 26th Apr, 10:31pm


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