THERE IS NO SPOON

Posted in Ed vs. Logical Fallacies on September 14th, 2011 by Ed

This isn't really political, but I did this in class on Tuesday and I had too much fun watching everyone tie their brains in knots not to share it here.

We are beginning to talk about probability in a course on research design / methodology, and I introduced the topic with a few basic examples. One of them is the infamous Monty Hall Problem. The MHP is like an optical illusion – no matter how many times I tell you that these lines are of identical length, your brain will keep telling you that the bottom line is longer. It is the kind of paradox with a solution that appears to be absurd even after it has been demonstrated to be correct. You simply cannot wrap your mind around it.

The MHP is a probability game named after the host of the 1970s/early 1980s game show Let's Make a Deal. The most popular part of the show was a game that has since become the subject of dozens of academic papers in math, statistics, and logic. Here is the basic setup:

The host confronts the contestant with three doors. Behind two doors there are goats (which the contestant did not actually win, but were intended as a gag "prize") while the third door hides a brand new car. The three prizes are distrubuted behind the doors randomly and Monty Hall knows what is behind each door. The contestant chooses a door, which remains closed. Hall must then open one of the other two doors to reveal a goat. He cannot open the door hiding the car (if the contestant has not chosen it). He then asks the contestant if they would like to stick with their original choice or switch to the other available mystery door.

An example is clearer; You choose Door 1. Hall opens Door 2 to reveal a goat. He then offers you the option of sticking with Door 1 or changing your choice to Door 3.

To 99.9% of humanity, this problem has an easy solution. Since Hall will open one of the two goat doors, the two unopened doors contain one goat and one car. There is no advantage to switching, since the odds of the chosen door containing the car appear to be 50-50. Going back to the example, if Door 2 is opened to reveal a goat, there's a 50% chance that Door 1 contains the car and a 50% chance that Door 3 contains it, right?

Switching, in fact, is always in the player's interest. When Parade magazine ran this problem in 1990 they received more than 10,000 critical letters, including several hundred from people with PhDs in math and science. All patiently explained that the odds are 50-50 and switching is not beneficial. The well educated letter writers were all wrong.

The probability of winning by staying with the original choice is 1/3, not 1/2. And the probability of winning by switching is 2/3. Most people, no matter how long they stare at this and work through the scenarios, cannot accept that. It took me days to absorb it when I first encountered the problem. It just does not make sense. Yet it's true. Since there are only two different objects behind the three doors – two goats and one car – there are only three possible combinations: car-goat-goat, goat-car-goat, and goat-goat-car. Perhaps the only way to understand why this creates a 2/3 probability of success by switching is to see the three scenarios spelled out. The yellow represents closed doors, the white door is the one opened by Hall to reveal a goat, and the arrow represents the door originally chosen by the player (which is always Door 1, for simplicity's sake).

And now that I've explained it and you've seen the irrefutable evidence, I bet your brain still doesn't comprehend why switching increases your odds. Mine doesn't. I know what the solution is but it will take me a few more years to understand why.

To paraphrase 50 Cent, I love this problem like a fat kid loves cake.