I can’t believe that this problem is still hanging around. The latest version is over at Dean’s World and reading the comments is…well painful. So, here is a formal proof that switching is always better than sticking.
First, there are three doors, Door 1, Door 2, and Door 3. Now for some notation:
Ci: Denotes the event that the care is behind door i.
Onot i: Denotes the event that Door not i is opened by Monty.
These are the relevant probabilities:
- P(Ci) = 1/3; this is the probability that the car is behind door i, where i = 1,2,3.
- P(Onot i) = 1/2; this is the probability that Monty opens door not i given that the care is behind door i.
- P(Onot i|Ci); this is the probability that Monty Opens door not i given that the car is behind door i.
Now what is the probability that the car is behind door i given that we are playing the strategy “stick” (i.e., we stick with our orignal choice) and that Monty has opened one of the doors we did not pick. That is,
P(Ci|Onot i) .
By Bayes Theorem we can write,
P(Ci|Onot i) = [P(Onot i|Ci) * P(Ci) ]/P(Onot i) .
Now, on the right hand side of the equation the first term in the numerator is 1/2 since the door is behind Ci Monty can pick either of the two doors not picked randomly. The probability in the numerator is also 1/2, and the “prior probability”–i.e. P(Ci)– is 1/3. Thus, the 1/2’s cancel out and we get a probability of 1/3. That is the Probability that the car is behind door i is 1/3 if we stick.
Now suppose we play “switch”.
We again use Bayes Theorem to come up with,
P(Ci|Onot i) = [P(Onot i|Ci) * P(Ci) ]/P(Onot i) .
Now we have chosen the door Monty did not open. Now, given that the car is behind the door we switched too, prior to that switch Monty’s only choice of door to open was the other door, hence the probability for opening not i is 1. The probability in the denominator is still 1/2, and the prior probability is 2/3.
Q.E.D.
Note, that in this proof I have not made any assumptions about which door was chosen by the player. Note also, that this is a mathematical proof. In other words, the switch strategy has been proven (i.e. 100% beyond doubt) to be the superior strategy. You can disagree if you want, but unless you can find a problem with this proof your arguments are wrong. Feel free to post any reason you think switching is no better than sticking, but unless you can show a flaw in the proof you are wrong.
Update: I thought I’d add some explanation at to why switching is the better strategy. When you first pick a door, you know nothing about each door. So when you pick chances are you picked wrong; after all your chance of getting the right door is 1/3 and the chance the other two doors have the car behind one of them is 2/3. Now, when Monty opens one of the doors you did not pick Monty is telling you something about the doors you did not pick, not about the door you did pick. In other words, Monty is telling you about the set of doors that is most likely to win, and switching takes advantage of that information while sticking ignores that new information.
If you are still not convinced consider a variant of the game. Suppose there are 1 million doors. Is your first choice likely wrong? The answer should be yes. Now after your first choice Monty opens up 999,998 doors that don’t have the car behind them that you didn’t pick. Now why is the door you first picked suddenly the likely winner? It is still has a probability of 1/1,000,000 of being the winning door. So the door that is still closed and you didn’t pick is probably the winner (with a probability of 99.9999%).
If you still aren’t convinced, then please…please, never go to Las Vegas or Atlantic city for anything other than the shows.
Update II: Damn it, but Rodney Dill, OTB contributor, has come up with a very elegant way of showing why switching is always superior. When Monty opens one of the two doors you didn’t pick and offers to let you switch to the un-opened door you are basically being allowed to pick both of the doors you didn’t pick the first time. Hence the probability of switching is indeed 2/3. Thanks Rodney.
