The day before the holiday break allowed for another fun and challenging exploration for my Fessenden students. This time we discussed the mathematical paradox known as the Monty Hall Problem from two perspectives, one mathematical and the other psychological.
Here is the scenario (based on a game show from the 80's),
Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, (who knows what’s behind the doors), opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?
The problem has confounded many experts despite its relatively basic probability scenario. I first heard of this paradox when many readers (some with Phd's) criticized a column written by Marlilyn vos Savant in the popular Sunday magazine,
Parade. You can read some of the harshly worded (yet wrong!) responses
here.
In the class before break, I become the game show host dressed in a gaudy sports coat and stick microphone and ask players to "come on down". We play the Pick a Door game using an
online simulator and students start winning fictitious goats and cars. The students eagerly play and also start to analyze strategies and patterns to improve their chances of winning. At first, patterns are elusive and most students believe that "staying" or "switching" shouldn't matter as it seems like an equal probability. We test their theories for many rounds before I show them that the simulator can also play automatically for 1000 rounds. This large number of trials show a pattern that becomes more and more clear -
Players will win 2 out of 3 times when switching but only 1 out of 3 times when sticking. This is puzzling to pretty much everyone in the class. My mathematical explanation to students:
- Pretend you always
stick no matter what. You will win 1 out of 3 times. This is the theoretical probability and will closely match the experimental probability with enough trials.
- Pretend you always s
witch no matter what. With this strategy, you only lose if you pick the door with the car first. If you pick a door with a goat, the host will always show you the other goat and you will switch to the car. The odds of picking a goat first is 2 out of 3.
The second part of our class is dedicated to the psychological analysis of how individuals play this game. I like this part even more than the first part! As stated before, most people don't think it matters if you stay or switch and put the probability at 50/50. In addition, some people take it further and ironically don't want to switch as they want to "trust their gut" and stay with their original choice. This was explored in depth by the television show Mythbusters in an episode called
Wheel of Misfortune. In class, we watch the episode and the results are astounding. 20 contestants play the Pick a Door game in an elaborate simulation and
every single one of them stick with their original door choice. Every one! Despite having twice the odds of winning, 20 people in a row make the absolute wrong choice and stick.
The conclusion I want my students to have is that you shouldn't always trust your gut. Be informed, use math to complete an analysis and take your time and hopefully you will make better choices in life.
