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the following is a quite well know thing but I personaly did not know of it (i think?) until reading the (most excellent) book "the curious incident of the dog in the night time" by Mark Haddon.

Anyway think of it as a brain teaser. If you already know of it then please do not spoil it for others.

You are on a game show

There are three closed doors. One has a car behind it and two do not.

The show host asks you chose one door.

After you have chosen one door the host then opens one of the door with nothing behind it that you did not chose, leaving two doors remaing one of which was your original choice.

The host then offers you a chance to change your choice of door from the one you originaly chose to the other remaing closed door that you did not

Should you stick with your original choice, change to the other door or does in not matter which you do?


The answer will be given in a week or so's time.

This is a mathematical based probability question, yes, you do change it because the player's chances of winning whatever it is are doubled when you switch, the solution is kind of counterintuitive, so it's a paradox.

I run it from the possibilities and indeed it seems that if you switch you have better chances.

However the chances are 50%-50% since no matter what you will chose the first time the second time you will end up with a door that has the car and with a door that doesn't. Basically the first time you have 1/3rd of chance of choosing the right door. The second time you have 50%. Your choice in the first time doesn't affect the possibility in the second time. The two times should be seen separately.

By the way, in my answer above I asume that the host doesn't know which door contains the car and that he randomly opens one other door. (this is what makes the two times separate)

If this is not the case (and the host opened a door that he knew beforehand that it doesn't contain the car), then you should switch.

I really don't see what difference it makes whether the host knows which door is the right one.
Even if we use probabilities, in the first instance the contestant's odds are 1/3 (33.3%) of hitting the right door, whereas in the second instance the odds are 1/2 (50%). That, however, does not mean that the contestant should change his or her initial choice. Both remaining doors have an equal 50% probability of being the right door. So, it really is a random choice, either way.

I've been thinking about this for two days, and I'm pretty sure the answer is not as simple as 'its just a random call and it makes no difference whether the contestant switches or not', but I just can't see it yet.

Unless it's one of those dumb 'hidden in plain sight' ones - the host presumably doesn't want you to win (a given) and knows you've chosen the right door. So he's trying to lure you into making the wrong choice (by switching to the other door), or else he'd be patting you on the back already - he wouldn't have given you another chance. In which case, you shouldn't switch.

Saint Jimmy wrote:I really don't see what difference it makes whether the host knows which door is the right one.
Even if we use probabilities, in the first instance the contestant's odds are 1/3 (33.3%) of hitting the right door, whereas in the second instance the odds are 1/2 (50%). That, however, does not mean that the contestant should change his or her initial choice. Both remaining doors have an equal 50% probability of being the right door. So, it really is a random choice, either way.

I've been thinking about this for two days, and I'm pretty sure the answer is not as simple as 'its just a random call and it makes no difference whether the contestant switches or not', but I just can't see it yet.

Unless it's one of those dumb 'hidden in plain sight' ones - the host presumably doesn't want you to win (a given) and knows you've chosen the right door. So he's trying to lure you into making the wrong choice (by switching to the other door), or else he'd be patting you on the back already - he wouldn't have given you another chance. In which case, you shouldn't switch.

i think your right

IM right, I study further Maths, I am a genius when it comes to these things.

1st pick odds are 3-1
one door with nothing behind it makes the odds for the 2nd pick 2-1

its 50-50 whichever door you pick.

ChomskyFan wrote:IM right, I study further Maths, I am a genius when it comes to these things.

further maths at what level. i am statrting my A-level maths in september. was going to do further maths too but then it would be too much work.

demetriou_74 wrote:

ChomskyFan wrote:IM right, I study further Maths, I am a genius when it comes to these things.

further maths at what level. i am statrting my A-level maths in september. was going to do further maths too but then it would be too much work.

I don't know, haven't decided what I am going to drop for A-Level yet.