Explain to me this probability puzzle

OK, a friend of mine told me this puzzle and I'm not sure I understand the answer. The riddle is:

You are on a game show on which there are three doors. Behind one is a car and behind the other two are goats. Only the game's host knows what's behind each door for sure. The host asks you to pick one and after you choose, he reveals which one of the doors has a goat behind it (not the door you have choosen, of course) and asks if you want to change your decision. What is the probability that you have picked the right door before and after he eliminates one door? Should you switch your choice?

My friend says that you should switch your choice after he eliminates one door, because the probability goes up that the door you did not choose is the one with the car. Since I am obviously too dense to get this, can someone explain this to me in small words?

Before the host eliminates one door, the probability of picking the car door is 1/3. After he tells you which door a goat is behind, it changes to 1/2. However, the only reason you would have to change your original choice is if the host told you that the door you'd already chosen had a goat behind it. Otherwise just stay where you are and hope you get lucky.

Your friend is wrong. The probablity that the door you didn't choose does go up, but so does the probablity that the door you chose originally does, too, to 1/2.

http://c2.com/cgi/wiki?MontyHallSolution

Read the short solution, at the top; it's correct. Read the rest if you want to see a very multiple-personality-looking discussion that results from people trying to converse by editing a wiki.

AHA! Now I get it. When I had first heard the puzzle, I thought the probability was 1/2 after the third door had been eliminated, but my friend had insisted it was 2/3 - he just couldn't explain why that was the case. This might make for a good puzzle in a game, if I can come up with an interesting and appropriate situation for it.

Thanks for your help!

Yeah, it depends on whether Monte knew which door the car was behind, and, if so, whether he is feeling malevolent, benevolent, or neutral.

If he didn't know, it was just luck that he picked an empty door. Your odds are 50/50 and you might as well stay put.

If he does know, there are three possibilities:

1. Maybe the only reason he is offering the choice is because you chose the correct door. (Evil Monte!). In this case you should stay put- your odds of winning are 0% if you change.

2. Or maybe he wants you to win, and is only giving you another chance because he knows you picked the wrong door. Since he opened the other wrong door, the only door left is the correct one- you should pick it. In this case your odds of winning are 100% if you change.

3. Or maybe he was going to give you another chance no matter what. In this case your odds are 66.67% in your favor if you change doors.

A friend of ours asked us the same question yesterday. I was the only one of the group that wasn't convinced that the solution was 2/3 in your favor if you changed doors. I'd just like to point out that I'm the one with a statistics degree :\ Nevertheless, I was finally convinced when I realized, that, yes, if you begin the game with the strategy "I choose a door and I'll change afterwards", you have a 2/3 chance of getting it right. But if your strategy is "random", ie pick a door at random, and then pick at random if you change or not, your probability becomes 1/2. That is all. Did I mention that I have a crappy computer that won't let me erase typos?

The other thing is (I think) if you don't know, at the beginning of the game, what the actual game will be (ie that you'll get a chance to change doors, or that they'll reveal a goat door), your chances are 1/2. Of course, maybe I'm wrong.

AR

eris404 said:

My friend says that you should switch your choice after he eliminates one door, because the probability goes up that the door you did not choose is the one with the car. Since I am obviously too dense to get this, can someone explain this to me in small words?

Click to expand...

Your friend knows absolutely dink about proper use of probability. Initial probability that you have chosen a car is 1/3. Probability after one goat is revealed is either zero or STILL 1/3. That one outcome is now known does not alter the fact that the chance of a car is one out of three total POSSIBLE outcomes.

Dogbrain said:

Your friend knows absolutely dink about proper use of probability. Initial probability that you have chosen a car is 1/3. Probability after one goat is revealed is either zero or STILL 1/3. That one outcome is now known does not alter the fact that the chance of a car is one out of three total POSSIBLE outcomes.

Click to expand...

Actually, the probability does change because you're given a second choice, and that choice is dependent on the first choice, making it a case of dependent probability.
Here's how it works. Let's number the doors 1, 2, and 3. Assume that 1 has the car (although you don't know that when choosing).

If you choose 1 first, and stay with 1 on your second choice (regardless of which door is eliminated), you win. One win for staying.
If you choose 1 first and switch to the other door not eliminated (regardless of which door is eliminated), you lose. One loss for switching.

If you choose 2 first and stay with 2 after 3 is eliminated, you lose. One loss for staying.
If you choose 2 first and switch to 1 after 3 is eliminated, you win. One win for switching.

If you choose 3 first and stay with 3 after 2 is elminated, you lose. That's two losses for staying.
If you choose 3 first and switch to 1 after 2 is eliminated, you win. That's two wins for switching.

Overall, staying wins one out of three times and loses two out of three times. Switching wins two out of three times and loses only once. So, you're better off switching (although it's not a guaranteed win).

As an addendum to my last post, I thought I should address arguments that the probability would be 50%. Once the first choice is made (1/3 of being correct), it's set in stone whether you chose right or wrong. If you made your next choice totally randomly, you would have a 1/2 chance of winning. However, knowing that your first choice (which by its nature, has to be totally random) was probably wrong, you can make your second choice not a random guess, and thus change to what is more likely the correct one.

Of course, there's still that 1/3 chance you're wrong.

2/3 if you change.

depends on how much of a gambler you are.

if you want to go with the safer bet. always change.

if you want to go with your gut... odds mean nothing.