Oldie but goodie

If this puzzle has been posted a thousand times please someone delete it.

You're in a game show, final round.

Behind one of three identical doors is a million dollars. Behind the other 2 there is nothing.

You make your pick and the host opens one of the doors you didn't pick showing you that it was empty. You now have the option of sticking with your original door or swapping to the other before they're both opened.

What do u do?

Originally posted by Marsan
If this puzzle has been posted a thousand times please someone delete it.

You're in a game show, final round.

Behind one of three identical doors is a million dollars. Behind the other 2 there is nothing.

You make your pick and the host opens one of the doors you didn't pick showing you that it was empty. You now have the option of sticking with your original door or swapping to the other before they're both opened.

What do u do?

Originally posted by FabianFnas
Do I recognize this problem - or not?
I think it is called the "Monty Hall Show' problem.
Seach the internet, or perhaps it's enough to search RHP Forums...

Originally posted by Marsan
If this puzzle has been posted a thousand times please someone delete it.

You're in a game show, final round.

Behind one of three identical doors is a million dollars. Behind the other 2 there is nothing.

You make your pick and the host opens one of the doors you didn't pick showing you that it was empty. You now have the option of sticking with your original door or swapping to the other before they're both opened.

What do u do?

It has, and it's debatable depending on which version you post.

In the version you've posted, we need to make an assumption:
Does the presenter want you to win the money?
Assuming that ratings are difficult to determine a correlation for, the answer to this is "no" because they don't want the show to lose money.

Now, he didn't say he was going to let you change beforehand, so he's introduced this aspect of the show AFTER you've picked one of the three doors.

Had you picked a door with nothing behind it, there's no good reason why he wouldn't let you go straight in and get the nothing, instead of giving you a second chance.

Instead, he's put you in a position where statistically there's more chance of it being in the other door (2/3 instead of 1/3).

So you've picked a door, and now the host is directly encouraging you to swap.

Thus you should stick.

Originally posted by doodinthemood
It has, and it's debatable depending on which version you post.

In the version you've posted, we need to make an assumption:
Does the presenter want you to win the money?
Assuming that ratings are difficult to determine a correlation for, the answer to this is "no" because they don't want the show to lose money.

Now, he didn't say he was going ...[text shortened]... or, and now the host is directly encouraging you to swap.

Thus you should stick.

I've checked the wikipedia page now, it does a good job of explaining why the statistical answer is switch: http://en.wikipedia.org/wiki/Monty_Hall_problem

Do check out the first "other host behaviours" though. I don't think enough space is given to the top one, given that to me it seems the situation most often presented, with no reason to assume the host does something every time.

Originally posted by doodinthemood
I've checked the wikipedia page now, it does a good job of explaining why the statistical answer is switch: http://en.wikipedia.org/wiki/Monty_Hall_problem

Do check out the first "other host behaviours" though. I don't think enough space is given to the top one, given that to me it seems the situation most often presented, with no reason to assume the host does something every time.

I would switch.
Before I was shown that one door was empty, I had a 33.3% chance of getting it right.

If I stick, I have a 50% chance.

If I change my mind, I have a 66.6% chance, due to Variable change.

Or something like that.

A variant of this problem is what led to the longest most hotly debated RHP thread ever, as I recall.

However, you have to assume some things before you answer.

1) Does the host know where the prize is when he has a door opened?

2) Was the host going to open a door and give you the option to switch?

3) Does the host have an interest in whether you win or not, and if so is he for you or against you?


Assuming the problem is the traditional Monty Hall problem, wherein the host does know where the prize is, and was going to give you the option to switch regardless of your initial choice (rendering the third question moot).. The answer is that switching doubles your chances.

The reason being is that it was inevitable the door opened was a losing door, as there is always one available to the host. There was never a chance you'd be staring at a prize knowing you'd picked wrongly at this point, so the chance you picked the right door is still 1/3.

In the 1/3 of the time we pick correctly, the 2nd door picked is empty 1/1 times.
In the 2/3 of the time we pick wrongly, the 2nd door picked is empty 1/1 times, because the choice is not random!
The 2nd door picked is always empty, and (1/3) / (1/1) = 1/3.

However, you do now know the opened door does not hold the prize, and so the remaining door will have the prize the other 2/3 of the time.


Now supposing the person (including the host) that chooses the unchosen door to open first had no knowledge of which door held the prize. In this case, it might be possible that the prize door gets opened, and you would walk way having lost because a swap wouldn't mean anything at this point. Now we are told that the door chosen in fact was empty.

Because it COULD have gone otherwise, our odds of having picked correctly goes up to 1/2. Switching or staying are equally profitable choices in this case.

In the 1/3 of the time we pick correctly, the 2nd door picked is empty 1/1 times.
In the 2/3 of the time we pick wrongly, the 2nd door picked is empty 1/2 times.
2/3 of the time the 2nd door picked is empty, and (1/3) / (2/3) = 1/2.


Now suppose the host knows where the prize is, but bases his choice of whether to offer a switch based on if we pick correctly. Presumably he will count on us to swap doors given the chance, and so whether we ought to swap given the chance depends very much on whether he wishes us to succeed to fail.

If he wishes us to fail, a chance to swap means we have picked correctly.
If he wishes us to win, a chance to swap means we have picked wrongly.

Here's a quick run-down of the answer (using geepamoogle's questions) without any justifications. Just take my word for it that these are all true.

1) Does the host know where the prize is when he has a door opened?

2) Was the host going to open a door and give you the option to switch?

3) Does the host have an interest in whether you win or not, and if so is he for you or against you?

1) If no, switch and stick both have 50% chance. If yes, look at question 2.
2) If yes, switch has 2/3 chance, stick has 1/3 chance. If no, look at question 3.
3) If he is for you winning, switch has 100% chance. If he is against you winning, stick has 100% chance. If he has no preference, but still only opens a door half the time, then there's 2/3 chance on switch and 1/3 chance on stick.

Originally posted by Marsan
What do u do?

Originally posted by Marsan
If this puzzle has been posted a thousand times please someone delete it.

You're in a game show, final round.

Behind one of three identical doors is a million dollars. Behind the other 2 there is nothing.

You make your pick and the host opens one of the doors you didn't pick showing you that it was empty. You now have the option of sticking with your original door or swapping to the other before they're both opened.

What do u do?

Originally posted by doodinthemood
Here's a quick run-down of the answer (using geepamoogle's questions) without any justifications. Just take my word for it that these are all true.

1) Does the host know where the prize is when he has a door opened?

2) Was the host going to open a door and give you the option to switch?

3) Does the host have an interest in whether you win or ...[text shortened]... l only opens a door half the time, then there's 2/3 chance on switch and 1/3 chance on stick.

Originally posted by geepamoogle
Now supposing the person (including the host) that chooses the unchosen door to open first had no knowledge of which door held the prize. In this case, it might be possible that the prize door gets opened, and you would walk way having lost because a swap wouldn't mean anything at this point. Now we are told that the door chosen in fact was empty.

B ...[text shortened]... ed correctly goes up to 1/2. Switching or staying are equally profitable choices in this case.