25 Apr 2005

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).

Monday, 25th April 2005, 8:20pm GMT

Filed under: Timewasters

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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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Monty and the Captive Goats

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