Famous brain teaser

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I saw this famous old brain teaser on a film called 21, with Kevin Spacey. The answer is easily found. But it was the explanation which drove me nuts.


You are a contestant on a game show.

There are three closed doors.

Behind one door is the star prize, which is a brand new car.

The other two doors have a booby prize behind them.

The game show host knows which prize is behind each door.

You are asked to pick one of the doors.

Instead of opening the door you have chosen, the host then opens one of the other doors. And behind this door is a booby prize.

So that leaves two remaining closed doors. Behind one of these doors is the second booby prize. And behind the other door is the car.

You are then asked whether you want to stay with your original choice of door. Or whether you want to switch to the other remaining door.

And the question is, whether you have the better chance of winning the car if you stay with your original choice? - or if you switch to the other door? - or whether each door actually has exactly the same chance of winning the car?
 
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Change your guess.
If you guessed right, it makes no difference
If you guessed wrong. the host will show you the wrong door, so

overall you increase your chances if you swap.
 
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When you get down to two possibilities then the odds are simply 50/50
It is either behind the door you picked or it isn't.
She was talking twaddle when she said the odds of 2/3 mean it is more likely to be behind the door you didn't pick.
Similar to Deal or No Deal. You pick a box, open it and you either win or lose. No skill or mathematical probability/possibility behind it.
 
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When you get down to two possibilities then the odds are simply 50/50
It is either behind the door you picked or it isn't.
She was talking twaddle when she said the odds of 2/3 mean it is more likely to be behind the door you didn't pick.
Similar to Deal or No Deal. You pick a box, open it and you either win or lose. No skill or mathematical probability/possibility behind it.

No, I'm sorry you are wrong, It's a well known problem, and your odds of winning increase if you change doors.

Here is a simulation, run it as many times as you wish, the odds are in your favour to change.

 
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For all those in doubt, cut some cards out, 2 zonks and 1 car, and 3 larger cards as doors.

Get some one else to run it, and keep rearranging the cards.

You can the be the contestant as many times as you wish, by changing your choice you will win the Car far more often.
 
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Surely it makes a big difference? If you guessed right originally, and you change your guess, you won't win the car?
No - you only have the host's action to go on.
Your choice is whether to swap to the other door or stick with the one you have chosen.
The host will show you an empty room either way.
 
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When you get down to two possibilities then the odds are simply 50/50
It is either behind the door you picked or it isn't.
She was talking twaddle when she said the odds of 2/3 mean it is more likely to be behind the door you didn't pick.
Similar to Deal or No Deal. You pick a box, open it and you either win or lose. No skill or mathematical probability/possibility behind it.
Nope, Try reading it again!

If you don't have the right door, the host will ALWAYS show you the empty room - he knows which the car is behind., so you aren't "just guessing".

2/3 of the time your first guess will be wrong, but then you effectively get shown which door it is... always change your guess.
 
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2/3 of the time your first guess will be wrong,

Yes you are right, 66% of the time, your first guess will be wrong.

And as the host always shows you a wrong door, 66% of the time the remaining door must be correct.
 
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Some people seem to understand this problem instinctively, like my friend, and some on here. But it was always really hard work for me, and I still only understand it in a very mechanical sense. So, for anyone else who has struggled, here is my foolproof way of understanding it. ;)

There is one crucial key to understanding the solution to this problem. The thing to remember is that the odds are set when the contestant makes their initial choice of door. And nothing that happens afterwards changes those odds.

So, when the contestant first chooses a door, there is a 1/3 chance that the car is behind that particular door. And there is a 2/3 chance that it is behind one of the other doors.

When the host opens a door to reveal a booby prize, we are left with two closed doors. The car is behind one of these doors. And the second booby prize is behind the other door.

But this hasn’t changed the original odds.

There is still a 1/3 chance that the car is behind the original choice of door. And there is still a 2/3 chance that it is behind one of the other doors. But we know it’s not behind the open door. So, there must be 2/3 chance that it is behind the other closed door.
 
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I like doing it with a million doors.

When you pick your door you have a 1 in a million chance that you've got the car. And a 999,999/1,000,000 chance it is one of the other doors.

You pick door number #327. At this point you can be virtually certain you don't have the right door.

The host then opens every other door except for door #711 showing the booby prize.

If you got your 1 in a million guess right then behind door number #327 is your new car. If you didn't guess right the first time, which let's face it you didn't, then it must be behind the one door the host deliberately didn't open.

This is a famous puzzle, mostly because a lot of people just don't get it. Including a lot of very clever mathematicians.
 
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