Saturday, June 6, 2020

Monty Hall Problem from a Movie

Time flies and it has been two years since my last post.  Somehow this important pastime has not seen much investment of my time in Hong Kong.

I was watching the movie 21 the other night, which was about a team of MIT students breaking bank at Vegas casinos.  The movie was entertaining as it has (some) action, romance, a twist, and most importantly, math, packed into two hours of lights-out.  I learnt how to improve your odds at Blackjack, by working in a team and assigning a count to a table based on the number of dealt high and low cards. 

The part that really caught my attention however, was when professor Rosa was teaching the Monty Hall problem through his interaction with his A-student and soon-to-be recruit, Ben Campbell.  This problem is often depicted in the form of a game show.  The grand prize, a car, is waiting behind one of three closed doors.  When the contestant picks Door A, the game show host, who knows exactly which door the car is behind, opens a door that is empty, say, Door C.  At this point, only doors A and B remain.  The host offers the contestant a chance to switch his pick to Door B.  Should he take the bait?  The answer is a definite yes.

Why?  Before any door is opened, the probability of the contestant picking the correct door is 1/3. No contention here.  This means that the probability of the car parked behind either of the other two doors (B and C) is 2/3.  So far so good. By opening Door C which leaves someone empty-handed, the probability of Door A being the winning door is still 1/3 because there is no change in information for this door.  However, when the empty Door C is opened, there is new information in the "other two doors" category.  The odds of 2/3 now goes solely to Door B. Hence your odds for switching from Door A to B have suddenly doubled!

Many people failed to comprehend the above when the solution was first presented.  My guess is most of your friends will probably not do the switch, even after you attempt to explain the math behind this - I have tried it on a few without success.  Anyway, please heed my advice if you do end up in a game show offering cars behind closed doors!


- PTS