SPOILER ALERT: Answers to the Monty Hall riddle!

SPOILER ALERT: Answers to the Monty Hall riddle!

Here is the answer to the Monty Hall/ Bertrand Box problem. There’s math for those of you who can translate numbers into english, and a nice picture for people who (like me) get tend to flip out when we see a large group of numbers. (I know your out there- be proud!)

Again, thanks to Cornell University. You can find more math and a nice, clear explanation (in words) as to why so many people get this wrong here :

http://www.math.cornell.edu/~mec/2008-2009/TianyiZheng/Conditional.html

The statement of this famous problem in Parade Magazine is as follows:

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, donkey. You pick a door, say No.1, and the host, who knows what’s behind the doors, opens another door, say No.3, which has a donkey. He then says to you, “Do you want to pick door No.2?” Is it to your advantage to switch your choice? (Whitaker 1990)

[spoi­ler]Many people argue that “the two unopened doors are the same so they each will contain the car with probability 1/2, and hence there is no point in switching.” As we will now show, this naive reasoning is incorrect. To compute the answer, we will suppose that the host always chooses to show you a donkey (When the problem and the solution appeared in Parade, approximately 10,000 readers, including nearly 1,000 with Ph.D.s, wrote to the magazine claiming the published solution was wrong. Some of the controversy was because the Parade version of the problem is technically ambiguous since it leaves certain aspects of the host’s behavior unstated, for example whether the host must open a door and must make the offer to switch).

Assuming that you have picked door No.1, there are 3 cases:

Notice that although it took a number of steps to compute this answer, it is “obvious”. When we picked one of the three doors initially we had probability 1/3 of picking the car, and since the host can always open a door with a donkey the new information does not change our chance of winning.

Basically, switching is to your advantage, resulting in you winning the car 2/3^rds of the time.