CH 2: Picking a Winner is as Easy as 1, 2, 3.
Considering all the problems covered in this book, the Monty Hall Problem has to be one of the most strikingly confounding ones and therefore it is an apropos first topic for discussion. The Monty Hall problem is a great example of how mathematics can sometimes be counter- intuitive to common sense. It is so named for the game show host, Monty Hall, who actually featured this problem on a real live game show.
This problem deals with probabilities. The typical set up involves three doors. The contestant (i.e. you) is told that behind two of the doors are two undesirable prizes, let’s say a desktop computer running a 20th century operating system and with minimum RAM. Behind the third door is a really desirable prize, say the Nissan GT-R sports car. Monty starts by asking which door you believe the Nissan is behind. You say Door One, joking that there can be only one prize. He then surprises you by opening Door Two revealing a giant clunky outdated computer. The audience lets out a gasp as Monty turns to you and asks whether you would like to switch to Door Three.
Now the question to you is whether you would increase your odds of winning that prized car by switching from Door One to Door Three. Most people incorrectly assume that both Door One and Three have the same probabilities of revealing the car. In actual fact, switching from Door One to Three is a wise move. You go from having a 1/3 chance of finding the car in Door 1 to a 2/3 chance
of finding the car with Door 3! Why, pray tell? Well, when you first were asked to pick a door, all three doors had the same chance of revealing the car. That means whichever door you chose, One, Two or Three, you have a guaranteed 1/3 chance of
getting the right door. Now, when Monty opened the surprise door, Door Two, and eliminated that door as an option for containing the car, you now are contending with only two doors where you are guaranteed to find the car. But as we just said before, your door, number one, is a 1/3 probability of being the correct door. Now, since we know that there is a 100% chance of it being either Door One or Door Three, and since we also know that Door One represents 1/3 of that probability, then we know that now that we only have one other door, Door Three, the remaining 2/3 must belong entirely to Door Three. So essentially, by revealing Door Two, we increased the probability of finding the prize behind the door you did not choose, Door Three. Now, it is fair to say that this is a rather counter-intuitive result, wouldn’t you agree?