A few years back there was a game show called â€œLetâ€™s Make A Dealâ€

One of the scenarios of the game show was similar to the following.

A contestant would try to win a prize behind one of three doors, numbered Door1, Door 2, and Door 3.

At first the contestant would guess Door 1. Then the moderator, I think his name was Monty Hall, would then say that there was no prize behind Door 3. Next Monty would ask if the contestant would like to change his
decision to Door 2.

Which of the following is the correct choice and why.

according to science, they are both equal, since both have the same amount of information describing them, none.

but if i were a contestant, and the announcer didnt usually say where there was no prize, i think id be inclined to stick with door 1, since its like hes trying to distract me into trying door 2. you know like hes depserately trying to get yuou to change your mind any way he can. not that i wouldnt just stick with it anyway, but id be slightly more inclined.

This question appeared in the â€œAsk Marilynâ€ section of the â€œParadeâ€ magazine that appears in many Sunday Papers a number of years ago. At the time, Marilyn Vos Savant had the highest IQ ever recorded. Dreamer24_7 wrote: â€œMarilyn Vos Savant has a 228 iq from what i’ve heardâ€. I believe that is correct. In her book, I forget itsâ€™ name, she says that she received the highest number of responses, to any article she had ever ran, with this question.

Assuming that you actually want the prize, the answer to the question is Door 2.

If you chose Door 2, then you chance of winning is 50%. If you stay with Door 1, your chance of winning is approximately 33%. zenoâ€™s chance is 0%.

This statement was disputed by many PhDâ€™s in Statistics. Eventually a number of statistics classes even ran simulations of this problem with the result stated in the above paragraph.

This seems like a truly remarkable statement. You have sealed your fate on your first guess. And you can not undo it, even if you are given a second chance at a later time (unless you chose a different option).

This question appeared in the â€œAsk Marilynâ€ section of the â€œParadeâ€ magazine that appears in many Sunday Papers a number of years ago. At the time, Marilyn Vos Savant had the highest IQ ever recorded. Dreamer24_7 wrote: â€œMarilyn Vos Savant has a 228 iq from what i’ve heardâ€. I believe that is correct. In her book, I forget itsâ€™ name, she says that she received the highest number of responses, to any article she had ever ran, with this question.

Assuming that you actually want the prize, the answer to the question is Door 2.

If you chose Door 2, then you chance of winning is 50%. If you stay with Door 1, your chance of winning is approximately 33%. zenoâ€™s chance is 0%.

This statement was disputed by many PhDâ€™s in Statistics. Eventually a number of statistics classes even ran simulations of this problem with the result stated in the above paragraph.

This seems like a truly remarkable statement. You have sealed your fate on your first guess. And you can not undo it, even if you are given a second chance at a later time (unless you chose a different option).

the way i understand the problem is this : there are 3 doors, only one holds a prize.

you pick one.

if the door you pick does not hold the prize, the host asks you if you want to switch. since he asks, your chances of winning anything if you dont switch are 0%. the chances to win something if you do switch are about 50%-50% except he is marginally more inclined to ask about a dead door.

is that the problem you proposed or did we diverge again ?

No, the host always asked if you wanted to switch, regardless of whether you picked a door that did not hold a prize. If he only offered a switch if he knew that the door you picked did not hold a prize, and tells you that door #3 did not hold a prize, it would make your choice pretty easy. The suspense of the moment would be pretty weak for the purposes of creating a dramatic moment on a game show, eh? Why not just hand them the prize and save the expense of having to pay a carpenter to make the doors?

The right choice is to change door. The chance now that behind this door is the prize is 50%, whereas the door you chose earlier is only 33%.

This illustrates a few things:

That knowledge improves your odds. In this case the knowledge is that you now know that one of the door has got no prize.

If the game presenter have not opened the door but just made one of the doors unavailable to choose, then there is no change in the odds, ie you no new knowledge.

Although this problem is a very simple one, it is very difficult for humans. Humans are not naturally wired to think probabilistically.

There many simulations of this game on the web. On of which is here, or a better one, here.

can he ask you do you want to switch to door 2 if door 3 held the prize ? if he cant, and when he asks the prize must be behind either the door he asks or the door you picked, the reason the odds are 33, 50, 0 is obvious. if he can, its strange.

zeno: Monty is a nice guy and doesnâ€™t lie-----much. So your statement about the obvious answer is correct.

The problem is that there are guys, like me, who have a hard time figuring our why, on the second chance, your odds are not 50/50. After all you have two choices in front of you and it is easy to think that this is a random distribution. I guess that, if Marilynâ€™s conclusion is correct, then the assumption of a random distribution, at that time, must be false.

At the very minimum chanbengchinâ€™s illustration number 3 is true for me.

abgrund: I have followed some of your posts and I must say I enjoy your intellect and general style of communication.

I believe that I have adequately described the setup, though I seem to have confused zeno, which is an oddity. The conclusion is straight out of Marilynâ€™s book, though I am doing it by memory (not real strong); and it seems to be supported by chanbengchin. This leaves me with my ending remarks (no place to hide here). Sorry to disappoint you - But I am curious.

That probability that the door your selected initially had the prize is 1/3.

The probability that the prize is behind the other two doors is 2/3.

Now one of the remaining two doors is opened and has no prize, but the probability of the prize behind one of these two doors is still 2/3.

And so the probability that the prize is behind the last remaining unselected door is 2/3.

Now if you had not chosen an initial door, and you are to choose a door only after one was opened showing you no prize, then the odds for you succeeding with either door is 50/50.

Now how your initial choice materially changed the uncertainty in the situation is explained as follows. Your choice limited the game host’s choice of doors to open, as he can only open the door without the prize. For 1/3 of the time, you may have selected the prized door and he is indifferent about flipping open either remaining door. But for 2/3 of the times you have selected a non-prized door and the game host is constrained to flip open only one door, ie 2/3 of the time the remaining unopened door is the prize.