Mathematics > Monkey Donkey Math

Monkey Donkey Math

Published: October 7, 2010

I ran into the Monty Hall problem today in class. Another professor was teaching and he used the problem to start off the day. It is the perfect example of a math problem with a counter intuitive solution.

The problem is simply this. You are playing the old game show "Lets make a deal" and you pick one of the three doors. One door has a car (good) thing, the other two have donkeys (bad things). You pick any door, say door 3. Monty opens another door, say door 2. Then he give you the option to switch doors to door number 1. Do you switch or keep the same door?

Intuitively, it looks like a coin flip. You are now down to two doors, 50-50 chance you have the right one. One should be as good as the other.

But this other professor says no, that the smartest person in America solved this problem and she said you have twice the chance of winning if you switch.

I frequently disagree with smart people and that was where I was on this. But I talked with the the other professor a bit and just wasn't getting his the alternate approach. But I mentioned that my brother says my logic is always flawless. My caveat to my brother's comment is if I start from the wrong premise, I reason my way to the wrong answer using flawless logic. So If I'm missing this one, it is because I'm looking at the problem the wrong way-I have the wrong premise.

That is where the Monkey in the title comes from. There is an old story of the Monkey Trap that made the rounds in business circles in the 90's. The story there was that Africans could hollow out a small hole in a coconut and put a couple grains of rice in the coconut and chain it to the ground. A monkey would come up, put his hand in the coconut, grab the rice and make a fist. By keeping a hold of the rice the monkey hand was trapped inside the coconut and the trapper could capture the monkey. The story was created to show business people about a value trap, how they had to let to go of idea that was killing them.

It was a good parable, but it was also complete bullshit. No monkey was ever trapped that way. But since there are no monkey trappers in the USA, we ate that story up. Even though the underlying story was BS, people could relate to the principle, it resonated.

That resonation was part of the problem here. Once a person makes a pick, they feel that choice is as good as any other choice, so flipping from door 3 to door 1 is pointless. Just stay the course and see how they did. To some degree, their first choice becomes the monkey trap. The problem they have is that they see one donkey door. When they see that they know that they didn't pick one of the wrong doors. At some level, they feel better that they didn't pick that door and hence their pick is the right one. It does become something of a value trap and they think they are even money to win on to the original pick.

To understand the problem, I had to keep the all three doors closed in my head. Then I saw the problem for what it was. Monty knows what door is correct. If he has the right door in his two doors, he opens the wrong door. If both his door are wrong, he opens either. His job is easy, but yours just got harder because you are led into the Monkey Trap.

The better way to look at it is this. You pick one door, Monty has two doors. He asks if you will swap your one door for his two doors. In that case you would certainly swap as his two doors will win more than your one door, in fact exactly twice as often over the long run.

What you have to do is realize that that is exactly the offer he is giving you. He offers you both his doors for your one. But at first you don't see that because the one door is open and appears to be off the table. But in fact he is offering you both his doors. You just have to get your hand out of your coconut and accept. You won't always win by doing this, but you just doubled the expected win percentage.

As stupid as it seems, once you figure this out, you let go of the rice.

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