David K. Israel
The Monty Hall Problem
by David K. Israel - July 12, 2007 - 3:54 AM

usehall.pngSome months ago, I had the honor of interviewing Monty Hall for a story I was writing. If you don’t know, Monty created and hosted one of TV’s longest-running game shows, Let’s Make a Deal. (non sequitur: I’m once again reminded of that hilarious exchange in Airplane II: Lloyd Bridges’s character: “If anyone has any ideas – anything at all – now is the time to speak up.” Jacob: “How about a game show like Hollywood Squares, but with kids? Gary Coleman could host.”)

So Monty Hall. He was one of those figures I remember from growing up – always on TV, someone you could trust to make you smile, an affable host with a pretty interesting game show in his command. The interview went fantastically well, and I learned a lot about not only the origins of the show (the network pulled the plug after seeing the pilot, but later threw it on the schedule as a last-minute replacement for another show that had bombed), but Mr. Hall, too.
mh-1975.jpg What I didn’t know when I was a kid growing up watching the show, but only discovered in doing research for the interview, was something known as the “Monty Hall Paradox.” (AKA “Monty Hall Problem”) In a nutshell, the paradox asks the question: Do the player’s chances of getting the car (behind door number 1, 2, or 3) increase by switching doors once a guess has been made (so technically it’s down to two doors at that point).

I’ll post the answer after the jump, but I’d be interested to know what you think before you click through.

theorum.pngInterestingly, the answer is always YES. From Wiki:

Once the host has opened a door, the car must be behind one of the two remaining doors. The player has no way to know which of these doors is the winning door, leading many people to assume that each door has an equal probability and to conclude that switching does not matter (Mueser and Granberg, 1999). This “equal probability” assumption, while being intuitively seductive, is incorrect. The player’s chances of winning the car actually double by switching to the door the host offers.

For a very complex answer to WHY this is so, check out the full Wiki article, including a discussion of Bayes’ theorem. If any of you recall my post on Mark Haddon’s novel, the curious incident of the dog in the night-time, and have read the novel, you’ll also find a pretty cool discussion of the Monty Hall Problem there.

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Comments (10)

  1. I was actually asked to work out this problem for a job interview for a software company. I had to draw out my steps on a whiteboard so guessing wouldn’t cut it.

    posted by Yonit on 7-12-2007 at 6:35 am

  2. I believe that the great mathematics puzzle maven, Martin Gardner, came up with this in one his Scientific American columns. I was thinking about this when watching “Deal or No Deal”. If you are lucky enough to get the final decision and have to choose between “your” case and the one on the stage, using the “Monty” principle you would have theo swap cases. 1 in 2 chances is far better than 1 in 26.

    posted by F.X. RYAN on 7-12-2007 at 6:44 am

  3. The trick for seeing this (or getting people to see it) is to imagine that there are 100 doors. You pick one, Monty eliminates 98 and leaves one, and says “Do you want to stick with your door, or switch and take the door we didn’t eliminate?” One of them has to have the car behind it — odds are 1 in 100 that yours is it, odds are 99 in 100 that it’s Monty’s. With three doors the numbers scale down, but it gives you an idea of how this works.

    posted by EV on 7-12-2007 at 6:52 am

  4. I LOVE Monty Hall…I’m so impressed that you got to interview him…I imagine he’s as nice in person as he was on television..i loved his show – ‘anyone have a hard boiled egg? I’ll give you $50 for a paper clip…who has a stapler?’…

    posted by donner on 7-12-2007 at 7:40 am

  5. He was even nicer in person than he was on the show, donner. He had some very funny stories about some of the folk in the audience who used to hang on to their $25 winnings, without trading up. And some great stories about getting whacked in the face, the leg, falling down on stage — all the result of some of the more outrageous costumes people would come in with!

    posted by David on 7-12-2007 at 8:05 am

  6. This is a great post! I’m only slightly disappointed you could write about Let’s Make a Deal but not mention the lovely Carol Merrill…

    posted by Lauren on 7-12-2007 at 9:46 am

  7. Right, the whole reason the answer to this problem seems statistically counterintuitive is the fact that the door Monty chooses to reveal to you is not a random choice. Monty knows which door has the prize, and which doors don’t. Once you choose a (random) door, Monty will then reveal to you a door that is guaranteed not to have the prize. At this point, your chances of success are 1 out of 3 with your original door (that hasn’t changed since your initial decision), and 2 out of 3 with the door NOT revealed by Monty. So switching after Monty takes away one (known to be a booby-prize) door doubles your chances of finding the prize!

    posted by Jef on 7-12-2007 at 11:35 am

  8. I read TCIOTDITN-T, and what has been bothering me about this is that, sure, switching doors improves your chances, but what if the car is behind your first door?

    posted by hal on 7-12-2007 at 1:03 pm

  9. One factor they miss in all those elegant analyses is whether Monty wants you to win or not. You see, sometimes Monty gives contestants a chance to switch doors, and sometimes he didn’t.

    If Monty knows you have picked the right door, and he wants you to win, he isn’t going to give you the opportunity to switch. If he wants you to lose, he gives you the opportunity to switch.

    If Monty knows you have picked the wrong door, and he wants you to lose, he isn’t going to give you the opportunity to switch. If he wants you to win, he gives you the opportunity to switch.

    Therefore, you have to figure out whether Monty is on your side. If he is, and he lets you change your mind, take it. If he isn’t, and he lets you change your mind, don’t.

    posted by Paul Ding on 7-12-2007 at 10:59 pm

  10. FX Ryan: It doesn’t work for deal Or No Deal. Swapping with Monty Hall works because he will always take away a dud prize. In DOND you have an equal chance of it being one of the two amounts left.

    posted by haribo on 7-13-2007 at 9:50 am

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