Pick a Door, Any Door

You’re trapped in a bland room with absolutely nothing but three ordinary, poster-plastered doors. The first door is covered with pictures of the teen pop star sensation, Justin Bieber singing in front of an audience of entirely teenage girls (roughly the composition of his entire fan base). The second door displays a montage of the cast of Modern Family, the TV show, which seems to have grasped the public’s short attention span for the time being. And lastly the third door includes a very sharp arrangement of Atlanta Falcon paraphernalia. You are told that behind one of these doors is a free ticket to any concert of your choosing, but behind the other two doors are free (did I mention enforced?) enrollment courses to…drum roll please…SAT classes! Your parents, who happen to spontaneously appear in your room, instruct you to choose any of the three doors and claim the “prize” behind it. Which door do you pick? Chances are if you value talented musical artistry, you’re at least not going to choose the Bieber door. Alright, alright, enough with the Bieber jokes. So you pick a door, the Bieber door. Now the catch is, your parents don’t open the door you chose. Instead, they open one of the other two doors (let’s just say the Modern Family door) and lo and behold, you get glimpse of the dreaded SAT class. Now, your parents offer you the once in a lifetime opportunity to either change your choice to the Falcons door or stick with Bieber. What do you do?

This problem is essentially the teenage equivalent of the Monty Hall Problem,which you may recognize from the movie 21 directed by Robert Luketic and starring Kevin Spacey, Jim Sturgess, and Kate Bosworth. In this movie, the professor Micky Rosa, played by Kevin Spacey, poses a perhaps less embellished version of this dilemma to a student Ben, played by Jim Sturgess. Ben, being thoroughly well-versed in the principles of probability and statistics, decides to change his door. Why? The truth is that this entire problem is designed to play into the decision maker’s emotional bias. My version of the problem may attempt to do this more so than other versions given the decals on the faces of the door, but what’s life without a little imagination? While the scope of the movie 21 incorporates more than just this 5-minute scene, I wanted to take the time to investigate the mathematics behind this choice.

When you first pick a door, there is approximately 33.3% chance (1/3) that your door contains the coveted concert tickets and about 66.7% that the doors you didn’t choose contain the tickets.

When your parents open one of the other doors to reveal the SAT class, how has the situation changed? Probability dictates that since your door has a 33.3% chance of containing the tickets, the remaining door now has a 66.7% chance of containing the tickets.
Thus, in reference to our earlier example, the conditional probability that the Falcons door contains the prize given that you originally chose the Bieber door and that the Modern Family door contains the SAT class is approximately 66.7%.

Still aren’t convinced? Try thinking about the problem in the context of 100 doors. Your original door has 1% chance of containing the prize while the other 99 have an added total of a 99% chance that one of them contains the prize. As 98 of those 99 doors are exposed as “phonies”, the last door standing has a 99% chance of containing the prize as it has managed to outlive a 98-door elimination process. Now if you still don’t believe me, I encourage you to try it out! Take three cups and put a ping-pong ball (or any quiet object) under one of the cups. Then have a friend guess which cup contains the “prize”. Open one of the other cups (one not containing the prize) and ask if he wants to switch. Consistently run this experiment using his original choice regarding switching. If he chose to switch, he should be winning about 2/3 of the time. If he chose to stay, he should be winning about 1/3 of the time. Stay tuned for future posts about Math & Movies!

Image Sources:

Justin Bieber: wikipedia.com

Modern Family: wikipedia.com

Atlanta Falcons: ticketmaster.com