Tue 6 May 2008
The Doors of Deception
Posted by anaglyph under Ephemera, Mathematics, Philosophy, Science, Skeptical Thinking
[19] Comments
A couple of days back I was reading an article about statistical method in experiments in primate behaviour and the writer mentioned The Monty Hall Problem as a possible source of unintentional introduced error or experimenter bias.
Now The Monty Hall Problem is a fascinating mathematical conundrum, and since I know those kinds of things are always of interest to Cow Readers, I thought those of you who are not familiar with this puzzle might like to exercise your mental muscles on it.
The Monty Hall Problem goes like this:
You are on a game show with your host Monty Hall who is offering you the chance to walk away with the Car of Your Dreams. He shows you three doors, A, B & C.
â€œThe Car of Your Dreams is behind one of these doors,â€ he says, â€œThe other two doors each conceal a goat. As your Games Master, only I know which door conceals which object. Now, please choose a door to claim your prize!â€™
You choose your door. You tell Monty â€œI have chosen Door B!â€
â€œWell done!â€ he says. â€œI knew you were a contestant of superior ability! But before we open your door, Iâ€™m going to open one of the other doors and show you whatâ€™s behind it.â€ He opens Door C to reveal a goat. â€œNow that youâ€™ve seen whatâ€™s behind Door C,â€ he says, â€œIâ€™m going to give you a special opportunity to stick with your chosen door, Door B, or change your choice to the other remaining door, Door A. I’ll give you ten seconds to have a think about it!â€
Hereâ€™s the question: To win the car, is there any advantage in changing your mind and swapping from your initial choice of Door B to Door A?
Answers on my desk by end of class.
19 Responses to “ The Doors of Deception ”
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Yes. You have a much greater chance of winning if you switch.
The reason being: You have a 2/3 chance of choosing a goat your first try. No matter what you choose, the host is not going to open your selected door. Instead, the host will have to select the OTHER goat in the other door. If you picked one of the goats and the host opens the door to the other goat, then switching means you get the car. Again, because the odds of you choosing the goat > than choosing the car, you’re much better off choosing a door with a goat the first time, allowing the host to reveal the other goat, then switching to the last door.
More importantly, though, why aren’t there cows behind the doors?
Cow Note: Atlas proposed a very compelling (and entirely correct) piece of reasoning to substantiate his decision, which I have excluded from comments until we’ve had some more thoughts.Comments restored as promised.
Oh, and there aren’t cows behind the doors because you have to have some incentive to try and get the car.
Choose a cow over a car? Udderly insane shenanigans!
As with all things to do with probablity, it depends.
In this case it is not entirely random as the Game host knows where the Car is, so his choice of opening a door is dependent on the contestants choice. This muddies the Probability waters. In the beginning. There is a 1 in 3 chance of selecting the car randomly, but when the choice is made and the host makes a choice it is influenced by his knowledge of the real situation, so this will need to be taken into account. I would say at this point that there is a 2 in 3 chance if you switch.
Buy yes why a goat?
S.
I’ll take the mystery box, and car is a car, but the box could be anything!
Peter Griffin moment there
The Monty Python Problem is very similar to this, except one door contains an African Swallow and one door a European Swallow.
Wait, do you win if you get the car or the goat? You can’t eat a car and the goat can’t carry you far. I’m confused.
And you have to switch. Because that’s what the darks powers of σ and μ decree.
This does actually make sense if you really REALLY think about it. Keep in mind, bourbon helps one understand the science of statistics.
Atlas: And the prize in that case is a Norwegian Blue Parrot.
Casey: Well, yes, there is a certain implication that if you’re a contestant on the Monty Hall Show you’re in favour of walking away with the car rather than a goat. If you’re in it for the goats then you are already ahead of the game.
I guess when you picked you only had a 33.sumin% chance of being right, but now the other door has a 50% chance of being right. (But so does your original 33.sumin% door.)
However, I hate hate hate regret more than anything. I avoid it at all costs. So, I would stay with the original choice because at least I wouldn’t be kicking my ass for changing my mind if the original choice held the car and I’d switched.
I guess making goat sounds in hopes the goats will answer and give you a directional hint is not a part of the equation here. Huh?
But the car of my dreams is a goat!
Plus they give milk – try geting that from a car!
It it that the statistical likelihood of it being Door A jumps to 50%, while Door B stays at 33%?
But I never change my mind once I’ve decided, so I would stick with Door B.
I think Buzzardbilly and I went to the same statistics class…
Um, and how is MHP a source of bias in gorillas?
If Monty opens a door and reveals a duck, what are the odds that its echo will quack? And if Monty is made of wood and he floats, does that make him a witch?
When I initially chose, I had a two in three chance of getting your goat. Even though the opened door reveals a goat, I still have a two in three chance of goating. But the unchosen/unopened door has a one in two chance of concealing a car. So I say – switch.
“I never change my mind once I’ve decided…”
Pil – try keeping to that at mah jong on saturday :)
sorry reverend, we know you’ll be there in spirit
I am in no mood to make any decisions today ;P
Guess what? Monty Hall is my cousin (on my father’s side, I think probably would be my third or fourth cousin not sure, hehee…)