Lets Make a Deal

The Donkey Behind Door Number Three

Let’s say you’re on Monte Hall’s “Let’s Make a Deal” gameshow. You get to pick a hidden prize behind door number one, door number two, or door number three. The trick is that two of the doors hide donkeys, and only one door hides the Maserati.

You pick door number one.

The host, in the countdown to your choice, shows you that door number three hides a donkey, and you gasp in relief. Aren’t you glad you didn’t pick that door?

Now Monte puts his arm around you in a confidential manner, and asks if you want to change your choice. Are you going to stick with door number one, or switch to door number two?

This is the question the Husband posed to me on our Date Night, and I said what many people say: “I’ll stick with my choice.” I figured I had a 50-50 chance of winning.

But the simple mathematical fact is that I’m more likely to win the car if I switch.

I had no idea people had been arguing about this for years. I can remember reading The Parade, which came as an insert in The Commercial Appeal, which was the newspaper my family subscribed to as I was growing up in Memphis, Tennessee. I mostly pulled out and read The Parade to catch up on the latest gossip about movie stars, but there was and still is a column called “Ask Marilyn,” where super-smart Marilyn vos Savant (who was in the Guinness Book of World Records as the person with the highest recorded IQ — 228) employed her exceedingly sturdy gray matter to solve problems that readers sent in.

And in September of 1990  her counsel in the 3-door situation was to switch your choice in order to increase your odds of winning. She received an estimated 10,000 letters from people telling her she was mistaken. But she wasn’t!

The Husband is a major math geek, and he’s reading a book called The Drunkard’s Walk: How Randomness Rules Our Lives, where he came across the “Let’s Make a Deal” scenario. After he told me about Marilyn and the switch, I picked up the book myself and devoured the third chapter where the explanation occurs.

And basically I failed to take into consideration the game show host. You see, Monte knows the answer. He’s not going to reveal the car, he’s going to show you a donkey. And thus he changes the odds for you.

At first you had a 1 in 3 chance of guessing right off where the car was. IF you were lucky and the car really IS behind door number one, you’ll obviously win if you stick, and the odds for that are 1 in 3. But if you guessed wrong and the car is behind door number two, you’ll win if you switch, and the odds for that are 2 in 3. That’s better odds, isn’t it? The strategy of switching is on average twice as successful as staying with the original choice.

I know, it’s confusing, and it took me a few times to get it, but once I did, I was pumped! You can play it out here and see for yourself how it works.

So. What the hell am I going to do with this miniscule tidbit of super-kewl information? I am going to be the megastar of the next party I go to, I’m telling you. Math nerds are red hot.

Um, aren’t they?

Image credit: Wikipedia

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9 Responses to The Donkey Behind Door Number Three

  1. Lance Bledsoe September 11, 2009 at 8:39 am #

    Yeah, my wife’s pretty smart for a jock.

    I find it especially fascinating that among the many thousands of irate letters Marilyn received about her answer to this problem were several hundred from math PhDs, who were especially angry. One of these was the well-known mathematician Paul Erdos, who apparently got even angrier when he was shown a formal mathematical proof of the solution. The problem really is a fascinating one, and highlights something Mlodinow points out in his book (The Drunkard’s Walk), which is that “our brains are just not wired to do probability problems very well.” (Even if we do have a PhD in math.)

  2. Mary Jo September 11, 2009 at 2:42 pm #

    But what if I’d rather have the donkey? Certainly that changes the odds. And is it better if I don’t mention I prefer the donkey or broadcast that news? And why doesn’t everybody want the donkey? We boarded two one winter when I was growing up, and the donkey sings sweetly at the break of day, which I think would add great joy to many people’s lives. I’m just sayin’.

    • Lisa Creech Bledsoe September 11, 2009 at 3:16 pm #

      Gosh, I didn’t really think of it this way.

      Ok, your situation is this: There are two donkeys and one car, so your odds of winning (a donkey) are 2 in 3. Good odds, those. So let’s say you pick door number one. Monte then opens door number three to reveal… A lovely sweetly-singing donkey — uh, oh! What if a car is behind your door?? That would be awful! So he asks you if you want to switch.

      Do you switch or stay?

  3. Sine Botchen September 11, 2009 at 4:26 pm #

    I dunno.. I’ve often found myself disagreeing with Ms. Vos Savant (is that really her last name??) but I suck at math so who am I to say…

    Are you better off playing the same slot machine all night or randomly going from machine to machine hoping to hit a lucky one? My strategy was to listen to the sound the coins made when they dropped into the box, a fuller box meant the machine hadn’t “payed out” in a while. It didn’t work. In all fairness I’ve only been in a casino maybe ten times at the most (including cruise ships, etc.) Now I just look for a machine that has the highest rate of return (in case I do win something, anything) and plan on spending $20-$40 as “entertainment” (especially if they are bring complementary drinks to me – they don’t do that on carinval cruises, btw.)
    .-= Sine Botchen´s last blog ..My Handwriting Sucks =-.

  4. Dave September 11, 2009 at 4:27 pm #

    My brain hurts already, and I’ve only just begun to dig into this problem. I know what I’ll be spending the entire weekend doing: Trying to get that damn Maserati.
    Question: Did Monte Hall ever actually give away a Maserati on Let’s Make a Deal?
    .-= Dave´s last blog ..Airwaves Yamaha looks forward to Croft =-.

  5. Sine Botchen September 12, 2009 at 8:39 pm #

    Regarding the car issue.. yes! they actually did give away cars to real people.. our neighbors were from California and they had a green chevy nova (or maybe it was a green ford pinto??) they won on Let’s Make a Deal.. the paint was really faded, which lended to the credibility that they could have possibly won it on a TV show. anyway, didn’t have any reason not to believe them.
    .-= Sine Botchen´s last blog ..Riding in the rain =-.

  6. Sine Botchen September 12, 2009 at 8:54 pm #

    re: buick electra.. my grandmother had one of those! when she finally had a stroke and had to be hospitalized my mom & me went to her house in Jackson Mississippi to clear out her stuff.. I was 15 at the time and got to drive that big old car all over Jackson (somehow being from Texas seemed to excuse me -at least in our minds- from any perceived traffic laws of the great state of Mississippi). We were there all summer and while I had been to Jackson at least once, if not twice, a year to visit my grandparents as a kid (along with my brother/sister/mom/dad) it was the first time I got to hang out with my uncle, aunt & cousins as a (nearly) full fledged adult. It was totally liberating to have my mom toss me the keys to that big old buick and be told to go grocery shopping (and to not actually get lost!). I only have to think back to those days to know how my friends who come to Austin feel when they actually start learning their way around town the first couple of times.
    .-= Sine Botchen´s last blog ..Riding in the rain =-.

Trackbacks/Pingbacks

  1. urbanmelodies.com - Book Review – The Drunkard’s Walk - September 14, 2010

    […] When I write a review of a math-related book, I generally assume that it will be largely ignored by anyone’s who’s not a math geek.  While that may often be true, in this case I happen to know for a fact that this book has appeal beyond the math geek crowd because after I mentioned Mlodinow’s description (in Chapter 3) of the controversy that surrounded a fairly well-known math problem (the Monty Hall Problem), my wife was so fascinated that not only did she read the chapter for herself, she actually wrote a blog post about it. […]

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