I think I've finally figured out the 'Monty Hall Problem'. It was mentioned in the book I introduced in my last post, "200% of Nothing". When I first heard it, I was skeptical. Then, I was absolutely against it. This post was actually intended to make it look ridiculous and point out the obvious. However, when I searched for a picture to put up with my lambasting of the problem, I found one that I was able to wrap my mind around.Â
If you're unfamiliar with the problem, here's a quick explanation. Behind three doors (of a game show, maybe) are a goat, another goat, and a car. You pick door 1, 2 or 3, which you believe hides the car. One door is opened, revealing a goat. Now, there are two doors: one with a goat, one with a car. Most people (like myself, until just now) believe the odds to be 50-50. However, mathematically, you are more likely to win if you switch doors.
Why? It doesn't make sense. One door hides a goat, the other a car. Switching should be irrelevant. I once drew this grid:
Situation 1. You picked the car: Switch = loss. Stay = win.
Situation 2. You picked the goat: Switch = win. Stay = loss.
Two wins, two losses. 2 wins out of 4 possible moves is 2/4 = 1/2 = 50%.
My logic would work if Situations 1 and 2 were equally likely. But, as you can clearly see from the image, since there are two goats and one car, you are more likely to have initially picked a goat. Therefore, Situation 2 is twice as likely to occur than Situation 1 (2/3 = 2 x 1/3). Therefore, it would be best to switch, as Situation 2 yields a winning scenario when the doors are switched.
If the experiment were tweaked so that there were 4 doors (2 cars, 2 goats), and 2 doors were opened (1 car, 1 goat), then the logic would dictate that switching is irrelevant, because the initial odds of picking a goat would be 50-50, not 2-1. Each situation would be on equal footing, unlike the 3-door scenario.Â
What's so very interesting and applicable about this is that I had this scenario presented to me at least four times before, in varying media (class lecture, animated video, written form). It's safe to say that each time the problem was presented, it was not presented incorrectly. Why, then, did it take me so many times to finally get it? Is it the quantity of demonstrations that did it? Possible, but I reread that portion of the book several times, and actually became more indignant at the 'foolishness' of the author. I believe it was the method of presentation that did the trick. I am a very visual person, so pictures and videos are usually the best way for me to learn something. The video I watched years ago probably didn't work because of how it was verbally explained (which may have been perfectly sensical to its creator). With the still image, I was able to stare at it for a few minutes and come to my own realization. You can't give someone learning. It must be internalized or 'wrapped around by the mind'. Once I could explain it in my own terms, I was able to learn.Â
This is why I find it critical to be able to explain a concept in a number of different ways so that students have multiple opportunities to internalize it. Math is very agreeable to this, because many concepts are able to approached from varying angles (pun intended). And, with my propensity to use metaphors, visuals, and images, I can help students of varying styles learn their own way.