The Three Door Puzzle

[JudgeJ]

Here’s a poser which I would be interested to see how people answer.

A game show consists of a star prize (e.g. a car) behind one of three closed doors, and booby prizes behind the other two. Contestants simply have to choose a door and they win whatever is behind it.

Now the tricky bit. After the contestant has made his selection, but before revealing if he has won the car, the host (who knows where the car is) opens one of the remaining two doors to reveal a booby prize. He then gives the contestant the opportunity to change his mind to the other unopened door if he so wishes.
The question is, armed with this new information, should the contestant now change his mind or stick with his original choice? Does changing his choice alter his chance of winning?

I know the answer (and the logic behind it) which I shall post in a few days. I have a wager going with some colleagues on the likely results of your replies. Don’t let me down!

[Aquariel]

I might be wrong, but here goes ... originally, the chance of winning is 1 in 3; the host opening one door improves the chances of winning to 1 in 2, or 50%.

So I think you want us to say that it doesn't matter which of the 2 remaining doors he chooses, his chances won't improve beyond 50% ...

[clarkeyboy]

Theory says you should always change your mind. your original choice had only a 1 in 3 chance of being correct but, having eliminated one door, changing your mind gives you a 50-50 chance of being correct.

I personally believe that simply making a conscious decision to either stick or change is the same as changing your mind.

[crofter]

Be careful, this poser has history! I met it as the Monty Hall Problem. Although I originally went for the 50-50 solution, I have seen a fairly convincing proof that argues for a switch - leading to a two-thirds chance of winning.

I look forward to seeing what JudgeJ has to say!

[pablito-uk]

Hi. Maybe you saw the question in this book. If you didn't I highly recommend that you read a copy:

'The Curious Incident of the Dog in the Night-time' by Mark Haddon.

It answers the question, as you also will soon. Look out for the book though. It'll stick in your mind for a long time! Cheers.

[lastresort]

You have an increased chance of winning if you change your mind amd choose the other door.

[coolplayer]

try visiting the following site before you answer:

www.comedia.com

[coolplayer]

sorry try this site:

www.comedia.com/hot/monty-answer

[coolplayer]

sorry I'll get it rigtht yet!

www.comedia.com/hot/monty

[coolplayer]

final go here goes again

www.comedia.com/hot/monty.html

[biley]

Well JudgeJ, don't really think you can be judged to have won (or lost) your wager on the few answers you have received here. What perhaps you did not realise is that, having asked your question, there will always be at least one person who will give the actual answer in one form or another. It also has to be said that the intelligence of people who respond to questions in this section of answerbank is undoubtedly greater than that of the general population.

That said I totally admit that I was prepared to go to war against those who said that changing choices in the instance you describe actually DOUBLES your chances of winning - the whole idea of that does seem preposterous (and I believe there were many people who were quite vehement in their opposition to the very idea). But when you take the problem a little further and have 100 doors to choose from and when 98 are opened and revealed as having only booby prizes, then it becomes much more obvious that one should change ones original choice - giving one not a one in a hundred chance of winning now but a ninety-nine in hundred chance.

Good game/puzzle/poser and one which will fool the general population time after time after time.

[JudgeJ]

Thanks to those of you that answered. As has been said, not very conclusive. I was convinced that the overwhelming response would be that it makes no difference. After all, the host revealing the prize behind one of the doors changed nothing. This was my initial reaction when I heard this poser some years ago.

Crofter has covered virtually all the important points, especially when expanding the problem to one with 100 doors of which 98 (i.e. two less than the total) are revealed.

The simple answer is that you can only lose by changing if you chose correctly in the first place. Since the chances of choosing correctly initially are one in three, the chance of winning by sticking with your original choice is also one in three. Hence there is a two in three chance of winning by changing. The important point to remember is that the overall probability of winning must still be expressed in terms of chances out of three, but the host has given you a “free go” by revealing one of the losers.

I’ll try to convince my colleagues with whom I had the wager to agree to a draw!

[Mrs Creosote]

Changing the odds from one to three to two to three seems to double the chances of winning as far as my calculator is concerned.